PARTIAL DIFFERENTIAL EQUATIONS (PDE)ocw.nctu.edu.tw/course/engineeringmathematicsII/ch06.pdf · partial differential equations (pde) 機械工程學系白明憲教授

PARTIAL DIFFERENTIAL EQUATIONS (PDE)

機械工程學系

白明憲

教授

MingsianMingsian R R BaiBai2

Physical systems (with ldquodistributedrdquo parameters) depend on not only the ldquotime variablerdquo but also the ldquospace variablerdquorarr Governing equations are PDErsquos cf ODE for ldquolumprdquo parameters 1 independent variable t

Modeling governing equations see examples in Sec 112

Define problem apply physical laws PDE solution

Vibrating string

( )( )

Assumptions 1 motions constrained in one plane plane

2 small deflection 3 equal tension at both ends4 small slope

y x tT

( ) f x y y tamp

T tension force

mass per unit length (density)( ) distributed force per unit length

sin sin | tan |Transverse force ( small slope)

sin sin | tan |

Newtons law F

x x x x

m xf x y y t

T T TT T T

α α αα α α

+Δ +Δ⎧⎨⎩

Δ = Δ

uarr + =

partΔ

tan | tan | ( )

tan | tan |( ) ( )

( 0) (tan ) ( )

( ) ( )

y T T f x y y t xt

y T f x y y tt x

x T f x y y tx

yT f x y y tx xyT f x y y t

α αρ

= minus + Δpart

minuspart = +part Δ

partΔ rarr = +partpart part= +part partpart= +part

( ) f x y y tamp

Thus we obtain the following wave equation (1D)1 ( )

where wave speed (1D wave equation)

Extension For a membrane one has the 2D wa

y T y f x y y

fρ ρ

part part

part partminuspart

=part part

2 2 2 22 2 2

2 2 2 2

2 2 2 2 22 2 2

2 2 2 2 2

ve equation ( )

Sound wave 3D wave equation ( )

z z x y tz z z zc c z

t x y tp p x y z t

p p p p pc c pt x y z t

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

part part part partminus + = minus nabla =part part part part

part part part part partminus + + = minus nabla =part part part part part

For the wave equation with 0

---------(1)

homogeneou

t ( ) ( )

y x t y

c ct t t

ζ ηζ η

ζ η ζ ηζ η

ζ η ζ η

ζ η= minus

rarrpart part part part part part part= + = +part part part part part part partpart part part

part part

part part part part= + = minus +part part part part part part

part part

drsquoAlembert solution and method of characteristics

2 22 22 2

y yc c y c c yx t

ζ η ζ η⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

part part part part part part= rArr + = minuspart part part partpart part

2ζpartpart

ηζ ηpartpart

part+ +part part

2y c⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2ζpartpart

ηζ ηpartpart

partminus +part part

y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

ζ η ζ ζ η ζ η

= minus +

part partrArr = rArr =part part part

rArr = + = +

rArr +int

Consider an string

initial displacement ( 0) ( ) initial velocity ( 0) ( )

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

y x t F x ct G x ctp x y x F x G x

part part= minusinfin lt lt infin lt lt infinpart part

part= =part

= minus + +there4 = = +Q

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

F x ct x ct G x ct x ctq x y xt x ct t x ct t

F x ct c G x ct c cF x cG x=

minusminusminusminusminusminusminusminusminus

part part minus part minus part + part += = +part part minus part part + part

prime prime prime prime= minus minus + + minus + minusminus

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

q x dx cF x cF cG x cG

cF x cG x cF cG q x dx

F x G x p x

F x G x F G q x dx

p xF x q xc

prime prime = minus + + minus

prime primerArr minus = minus minus minusminusminusminusminus

prime primeminus = minus minus

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

y x t F x ct G x ctp x ct F Gq x dx

cminus

minusprime+ minusminusminusminusminus

minusprime prime= + minus minusminusminusminusminus

= minus + +

minus minusprime prime= minus +

( ) 1 (0) (0) ( )2 2 2

x ctp x ct F Gq x dxc

+ minusprime prime+ minusint

Thus ( ) ( ) 1( ) ( )2 2

p x ct p x cty x t q x dxc

minus + + prime prime= + int(drsquoAlembert solution)Ex ( ) 0

( ) ( ) ( )2

p x ct p x cty x trarr larr

minus + +=P

( )p x

The general waveform is a superposition of two waves traveling in oppositedirections with the constant speed c (phase speed) Without any energy loss the wave is ideally lsquonon-dispersiversquo meaning the waveform would not change as it travels throughout the time

Classification of second-order PDE

are constants

Try the solution form ( )( ) 2 ( ) ( ) 0

( 2 ) ( ) 0 2 0 for nontrivi

u f x yAf x y B f x y C f x yA B C f x y C B A u

u u uA B Cx x y y

λλ λ λ λ λλ λ λ λ λ

part part part+ +

= +primeprime primeprime primeprime+ + + + + =

primeprime+ + + = hArr

=part part part part

Δ =2 2

if 0 eg 0 (wave equation)

if 0 eg 0 (heat equation

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

minuspart partΔ gt minus =part partpart partΔ = minus =part partpart partΔ lt +part part 2 0 (Laplace equation)

1 1 2 2

1 1 2 22

characteriLet and be two roots of 2 0 and are termed the Note 1 1

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

prime prime= minus = minus

minus minusprime primerArr minus + = minus +

Thus the characteristics satisfy the following ODE ( ) 2 0

( 2 ) 0 12i i

A y By C

A B C iλ λλ

prime primeminus + =

+ += = =

This should give you two independent solutions

Wylie p696Consider the equation

( ) 2 ( ) ( ) ( )

Let ( ) and ( ) be two indepen

(More generally are f

dent sol

ions of ( ) )

u u u u uA x y B x y C x y g u x yx

A B C x

x y y x yx y c x y c

part part part part part+ + =part part part part part part= =

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

dyA x y y B x y y C x y ydx

u u ur x y s x y G u r sr s r s

prime prime primeminus + = =

part part part= = rarr =part part part part

b ( ) ( )

( ) ( ) (

) ( )c 2 2

u u ur x s x y G u r sr r s

x y x y x y x yr si

u u u uG u r sr s r s

φ ψ φ ψ

part part part= = rarr =part part part

+ minus= =

part part part partrarr + =part part part part

Ex 0 Find the solution

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

A y By Cx y yy y xy y

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

part part partminus + =part part part part

= = minus rArr Δ = minus = gt

prime primeminus + =prime prime prime prime+ = rArr + =

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

y c y cdy dx y x cy x

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

u u u u ry y yx x s x s r s x

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

part part part+ =part part part

part part part part part part part= = =part part part part part part part part

2 2 2 2 2

Substituting the above expressions into the given PDE leads to

u s uys x s

u u u u r u s u u uy y y xyy x y s s r s y s y s r s s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

part part part part part part part part part part part= = + + = + +part part part part part part part part part part part part part part

partpart

uyspartminuspart

2u uy xy

r s spart part+ +part part part

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

u r s f r g su x y f y g xy

part partrArr minus = =part part part part

rArr = +rArr = +

2 0u u ux yx x y xpart part partminus + =part part part part

Separation of Variables

By using this method PDE ODEsFor example t can be described bythe followi

where is angular displacement

ionally vwave equatio

ibrating shaftn

a x lt xEa

part part= le lepart part

shear modulus mass per unit volume

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

(0 ) ( ) 0t l tθ θ= =Fixed-fixed

Fixed-free

Free-free (0 ) ( ) 0t l tx xθ θpart part

= =part part

(0 ) ( ) 0t l txθθ part

= =part

0 lBCs

Trial solution ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

at xX x T t a X x T tT t X xaT t X x

T t T t X x X xa

θθ θ

=part part=part part

primeprime primeprimerArr =primeprime primeprimerArr = =

rArr = =

Let 0 without loss of generality

( ) ( ) ( ) ( ) ( ) ( )( ) Not periodic in time

physically impo

t t x a x a

x a x a t t

T T X Xa

T t Ae Be X x Ce Dex t X x T t Ce De Ae Be

λ λ λ λ

μ λλ

plusmn

minus minus

primeprime primeprime= =

rArr = + = += + +

Q 對稱

0 0 ( ) ( ) ( ) ( ) ( ) ( )( ) Not periodic in time physically impossibl

( ) ( ) ( ) ( )

( ) cos

T XT t At B X x Cx D

x t X x T t Cx D At B

T t T t X x X xa

T t A t B t X x

μλλ

primeprime primeprime= =rArr = + = +rArr = = + +rarr

= minus gt

primeprime primeprime= minus =

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

2 Sinu motion in time soidal frequency periwith od

C x D xa a

x t X x T t C x D x A t B ta a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

Determination of unknown parameters For example for ends BC (0 ) ( ) 0BC1 (0 ) 0 ( c

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

BCs C D ICs A Bt l t

t C A t B tA B x t trial

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

= rArr + == = equiv rArr

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

Thus sin 0 or ie 1 23

A t B t

l t D l A t B ta

A B x t trivialD x t trivia

lll n n

λθ λ λ

λ λ λπ

equiv = +

= = equiv rarr= rarr

equiv = =

sin ) ( ) ( ) ( cos sin )

periodic in time are satisfied

nn n n n n n nx

an ax t X x T t A t B t

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

(前面已限制λgt0)(eigenfunction) (Dn is absorbed)

Note that no single ( ) can satisfy the arbitrary

( 0) ( ) and ( 0) ( )

However we hope the infinite sum ( ) which also satisfies

the BCs will satisfy the given arbit

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

IC1 (initial displacement)

( 0) ( ) si

Fourier

half-range that approximates si ( ) over ne serie (s 0 )

n n nn n

n x n at n atx t x t A Bl l l

n xx f x Al

π π πθ θ

infin infin

sum sum

2 ( )sin (see Thm2 74)l n xf x dx

l lπ= int 章節

Fourier si

IC2 (initial velocity)

( ) sin sin cos

( 0) ( ) sin

half-range that approximates ( ) over (0 )

ne series

n x n at n at n ax t A Bt l l l l

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

part = minus +partpart = =partrarr

2 ( )sin

2or ( )sin

Up to this point ( ) is completely determined

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n at n atx t A Bl l l

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

int( )ilar reasoning can be applied to the other see p495-502BC s

Orthogonal Expansion and the Sturm-Liouville Theory

eigenvalueseigenfuncti

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

Recall the example of the torsionally vibrating shaft

ICs expansion Fourier sine seriess( in sin)

x n xa l

⎫⎪⎪⎬⎪⎪⎭

Sturm - Liouville TheoryFourier coefficients are obtained based on

Not a coincidence Fourier sincos series is not the only way of expansion Other orthogonal func

orthogona

tions Be

rarr Q

Legendre Chebyshev

Ex 1-D Boundary Value Problem (BVP) 0 (0 ) BC (0) ( ) 0 (nontrivial solution eigenvalue problem)

y y x ly y lλprimeprime+ = lt lt

= =rarr

cos sin if 0The general solution ( ) if 0

If 0 applying (0) 0

0 ( ) 0 trivial 0 is not an eigenvalue( ) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

rArr = = rArr equiv rArr == = +

If 0 applying (0) 0

sin 0 (characteristic equation)( ) 0 cos sin

If 0 then ( ) 0 trivial solution

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

rArr == = += equiv rArr

If sin 0 123( 0 0 trivial)

( ) sin ie ( ) sin

0 ( ) ( ) functionorthogona

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

matriNot a coincidence Sturm-Liouville theorem (SLT)

finite ( ) algebraic ifunction nf

xEigenvalue pro

inite ( ) transceblem

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Elements of an eigenvalue problem finite boundary and homogeneous problem (homogeneous equation and boundary conditions)

Greenberg p497

( ) ( ) ( ) 0 ( )BC ( ) ( ) 0 ( ) ( ) 0 where are continuous on [ ]

( ) 0 ( ) 0 on [ ] are o

p x y q x y r x y a x by a y a y b y ba b p p q r a b

p x r x a b

λα β γ δ

prime⎡ ⎤⎣ ⎦prime + + = lt ltprime prime+ = + =

primelt infingt gt

Sturm - Liouville Problem

both 0 are not both zero a b p q rγ δ α β γ δ isin

(function space)

where is complex conjugate ( ) is the weight function

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Orthogonality 0 orthogonal

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

u v u x v x r x dx u v

α α α

= hArr

intint

Define the differential operator

The SLP 0 can be rewri

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

0 (Lagranges identity) = ie The operator of self-adjoint is

Lu v u LvL SLP

Thmcf Hermitian matrix = HA A

pf 1 = ( ) [( ) ]

where are twice-differentiable and satisfy at ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0 ( ) ( ) 0

a aLu v pu qu vr dx pu qu v dx

ru v BCs x a b

u a u a v a v au b u b v b v b

α β α βγ δ γ δ

⎡ ⎤⎣ ⎦minus prime prime prime prime+ = minus +

=prime prime+ = + =prime prime+ = + =

int int

Integration by parts

( ) ( )

[ ( )] [( ) ]

bLu v pu v pv u quv dx

p uv u v pv qv u dx

⎡ ⎤⎣ ⎦prime prime prime= minus + minus

prime prime prime= minus minus +

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

⎡ ⎤⎛ ⎞prime prime prime primeminus minus minus⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

prime prime+ = + =ne

prime primerArr minus = =

=prime prime prime prime prime= rArr = = rArr minus =

prime primeminus equiv

larly ( ) 01Thus 0 [( ) ] [( ) ]

Note the self-adjointness hinges on not only the operator but also on the

uv u v

Lu v pv qv u dx pv qv ur dxr

u Lv r dx u Lv

=prime primeminus equivminusprime prime prime prime= minus + = +

lowast

int int

BCs (cf A hermitian matrix has only to do with the operator)27

Thm 1 All of the SLP are

Pf (also see Wylies pf p514 Wronskin)

(1) (2) ( )

eigenvalues r

φ λφ

φ φ λφ φ λ φ φ λ φ

φ φ φ λφ λ φ φ λ φ

φ φ φ φ λ λ φ

minus minus = minus

) ( Lagrange identity)

0 (eigenfunction must be nontrivial)

0 ie QED

λ λ λ

there4 minus = isin

Thm 2 corresponding to distinct eigenvalues of the SLP are Pf Let and be two distinct eigenvalues

orthogEig

j j j k k k

j j j j jk k k

λ λφ λ φ φ λ φ

φ φ φ λ φ λ φ φ

= = (1)

(2) (1) ( )

0 ( ) ( Lagrange identity)

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

φ φ λ φ φ λ φ φ

φ φ φ φ λ λ φ φ

λ λ φ φ

λ λ λ λ

φ φ φ

minus minus minus minusminus

= = minusminusminusminusminus

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

Thm 3 Eigenvalues are simple (no multiple values)Thm 4 (Generalized Fourier Series) Let ( ) be the eigenfunctions of the SLP and ( ) and

Eigenfunction Expans( ) be

piecewise continuou

ionn x f x f xφ prime

s on Then ( ) constitute an (close and comp

(lete Wylie) and

Pf Let ( ) ( )

orthogonal basi) ( ) ( )

m n n m n n m m m mn

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

(generalized Fourier coefficient)

At this series converges to the mean (non-uniform convergence)

φ φφ φ φ

rArr = =

Many kinds of eg trigonometric function Bessel function Legendre Hermite Chebyshev etc all fall into the class of Sturm-Liouville Proble orth

ial fun

alitm ho

lowast

For the previous 1D wave equation 0 (0 ) (0) ( ) 0 ( )

( ) sin for 123

Comparing the problem above with the SLP ( ) + 0 or

y y x l st y y l BCn n xx n

py qy ry py

λπ πλ φ

primeprime+ = lt lt = =

rArr = = =

prime prime primeprime+ = + 0 ( ) ( ) 0 ( ) ( ) 0 one has 1 0 1 0 1 0 standard SLP

p y qy ryBC y a y a y b y b

p q r a b l

λα β γ δ

α γ β δ

prime prime+ + =prime prime+ = + =

= = = = = = = = = rarr

Thus (Thm1) are simple (Thm3)

0 ( ) ( ) sin sin (Thm2)

π πφ φ φ φ⎧⎪⎨⎪⎩

ne= = =

=int int

b lm n m na

m nm x n xx x r dx dx ll l m n

To show eigenfunction expansion in Thm 4 for example ( )

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

Half-range expansion of Fourier-sine series

l n xn l

+infin

minus=

( )f x( )f x x=

nonuniformconvergence

Ex Bending Vibration of a Beam - Modal Analysis systematic

( ) ()

st BC Simply supported

(0 ) 0 (0 ) 0

part part+ = minusminusminusminusminuspart part

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

=part part+ =part part

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

g a g g gg t C t C t

=gt SHM Eigen-value prob 0 =gt ( ) cos sin cosh sinh

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

φ β φ φ β β β β

minus = = + + +

part primeprime= = = =partpart= =part

x A x B x C x D xuu t g t t g t

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

primeprime= =primeprime= =

l t l g t

(4) (3)4 4

is not exactly SL operator But it can be proved

to be a self-adjoint operator

pf ( )

d x Ldx

d u d uLu v u vdx v u v u v uvdx dx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

primeprime prime primeprime primeprime= minus = minus + minusint

is self adjoint ( ) ( )

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

u v v u u Lv v u vu dx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

primeprime primeprime= minus

= minus

primeprime primeprimerarr rarr = minus

= rArr isin perp

= = + minusminus

primeprime = = minus +

int int

代入

cosh ( ) (2)x

x A Cβ β== minus minus minusminus

From (1) (2) 0 ( ) sin sinh ( ) 0 sin sinh (3) ( ) 0 sin sinh (4)From (3) (4) 0 ( (3) (4) 0 sinh 0)

φ β βφ β βφ β β β β

= == += = + minusminusminusminus

primeprime = = minus + minusminusminusminus= + = neQ

A Cx B x D xl B l D ll B l D l

sin 0 12

(natrual freq) (e-value)

2 natural modes ( ) sin (mode shape) (e-func)

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

1 normalizationφ =n1=n

(4) 4 440 0

Orthogonality It can be shown that

Eigenfuction expansion ( ) ( ) ( )

(modal analysis

φ φ φ φ δ

φφ φ φ φ β φ β δ

=infin

int int

sum ampamp代入

lm n m n mn

l lnm n m m n n m mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

(0) ICs is modal space for ( ) (0)

Complete solution ( ) ( ) ( )

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

Non-homogeneous Problemnonzero equation andor BCrsquos

2 21 2

0 st 0 st

2-D Poissons Eq st on a rectangular boundary

+ (Laplace equation)

Satisfies (1) (2)01 Laplace equation wi

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

part partnabla = + =

rarrnabla =

1 2 3 4

th non-homogeneous BCs

For example a 2-D problem (Cartesian coordinates)

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

st u y g yu L y g yu x f xu x H f x

u u u u u

⎫⎪⎪⎬⎪⎪⎭

part part+ =part part

= + + + (We need BCs to yield eigenhomogeneo functus ions)1u f=

1u g=2u g=

2 0unabla =

Homogeneous problem (equation BCs) FiEigenfunc nite bountion s daryrarr

2 24 4

Solving one of the sub-problems as follows

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

u ux yu y g y u L yu x u x H

= == =

Separation of variables ( ) ( ) ( )( ) 0 ( 0) 0 ( ) 0

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

rArr = = =primeprime primeprime+ =

primeprime minus =rArr

primeprime + =

u x y h x yu L y u x u x H

h L Hy h x h x y

h x yh x y

h x h xy y

n ( only homogeneous BCs can yield eigenfunctions)( ) ( ) 0 (1D SLT problem)

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

In the -direction ( ) ( ) 0 st ( ) 0cosh

( ) cosh sinhsinh 2

sinh( ) 0 cosh sinh 0

sinh( ) cosh sinh

x h x h x h Lzn n e eh x a x a xzH H

n Ln n Hh L a L a L a anH H LH

n L n nHh x a x a xn H HLH

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

primeprime minus = =plusmn= + =

= rArr + = rArr = minus

rArr = minus +

Let ( ) sinhcosh

Thus ( ) ( ) ( ) sin sinh

sinhcosh

n n nn n

a na h x a x Ln HLH

n y nu x y A y h x A x LH H

a n x Ln HLH

π πφ

infin infin

prime prime= rArr = minus

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

Fourier-sine series in 2sinh ( )sin

2 ( )sinsinh

Similar procedure can be applied to fund

u y g yn y nA L g yH H

yn n yA L g y dyH H H

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

2 st 0 on the boundary

nabla ==

= nabla = ==

u u Q uu

X dependent eigenfunction sin( )

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

nabla = rArr minus =

rArr minus

x n x Ln xu x y b yL

d b n x n n xu Q b Qdy L L L

d b n n xbdy L L

st (0) 0 ( )variation of par

( ) ( ) (

) sinh ( )sinh sinh (

2Fourier-sine series ( )sin ( )

π πξ π π ξξ ξ ξ ξ

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

n H y n n y n Hb y q d q dL L L L

d b n n xb Q x y dx q ydy L L L

(Differentiation is allowed here because both and sin satisfy homo )n xu BC

Alternative method 2-D eigenfunction p 273-275 Haberman st 0

Consider the related 2D eigenfunctions ( )( ) ( ) st ( ) 0 on the boundary

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

Substitution of the above equation into Poissons equation yields the following eige

nfunction expansion

mn mn mnn m n m

n x n yx yL H

n x n yu x y b x y bL H

π πφinfin infin infin infin

= = = =

= =sumsum sumsum

( ) ( )1 1 1

sin sin or ( )mn mn mn mn mnn n m

n x n yb Q b x y QL Hπ πλ λ φ

infin infin infin

= = =minus = minus =sumsum sumsum

The Laplacian can be evaluated by term-by-term differentiation since both and satisfy the same homogeneous boundary conditions This is a generalized Fourier series The eigenfunctions are ort

hogonal in a 2D sense

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

Q n x L n y H dx dyb

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

Note Sometimes boundedness and periodicity can be used as BCs too

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

Midterm Exam GH 513 (Wed)Vector calculusPDE

Eigenfunction expansion for non-homogeneous problems

Greens identity and eigenfunction expancf Greens fun

sion ction method

non-separable

BC (0 ) ( ) ( ) ( ) (moving boundary)

IC ( 0) ( )

( 0) ( )

Non-homogeneous BCs(1) Subtracting reference function(2) Eigenfunction expansion amp Gr

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

part part= +part part

=part =part

lowast

eens identity

2 2 Finite boundary2

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

u t u L tu x t x b tb c b

x b tx c b t

φφ φφ λφ

part partminus = rarrpart part

primeprimerArr minus =

primeprimerArr = = minus

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

orthogona( ) sin lit (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

Eigenfunction expansion ( ) ( ) ( ) rdinatesm

d b tu ux c Q x tt dt x

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part part= = +part part

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

uc x dxd b xq tdt x dx

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

Problematic differentiation

= cannot be justified at 0 since ( ) satisfies the homogeneous BC while ( ) does not The term - by - term

of ( ) wrt is not justifie

differentiatio d since ( ) and ( ) do not

x L xu x t

u x t x u x t x

2 2 21 1

satisfy the same BC ie ( ) ( ) ( ) ( )

(only when is satisfuniform convergenc ied differentiation and summation are interchang

homoge

nn n n

d xu b t x b tx x dx

φφinfin infin

part part= nepart part sum sum

以方波為反例

Key to undo the difficulty 1-D (proved by integration by p

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

partpart part part= minuspart part part part

part part partpart= minuspart part part part

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

φ φ part partpart part⎛ ⎞rArr⎛ ⎞ ⎛ ⎞ = minus + minus⎜ ⎟part part⎜ part part⎝ ⎠

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

part partpart partminus = minuspart part

part part primeprime=part part

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

minuspartpart minuspart part

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

partpart minuspart part

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

uc x dx c nx q t c b t A t B tL Lx dx

φ πλφ

⎡ ⎤⎣ ⎦

partpart = minus + minus minus

intint

This leads to a time-domain initial value problem decsribed by the ODE( ) 2+ ( )= ( ) ( ) ( )( 1)

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

d b t c nc b t q t A t B tdt L

u x f x b x

φinfin

⎡ ⎤⎣ ⎦+ minus minus

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

Find ( ) from the ODE in Eq(1) subject to the ICs in

n n n Ln n

f x x dxb

g x x dxu x g x b x bt x dx

there4 =

part prime prime= = there4 =part

intint

intsumint

qs (2) and (3)

Finally we arrive at the solution ( ) ( ) ( )n nn

u x t b t xφinfin

No non-uniform convergence problem)

Ex approach for solving BVPs

(Poisson equation) st ( ) ( ) on

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

function (time-domain Greens func)

position vector of the position vector of the By Greens 2 identit

response

y[ ( ) ( ) (

field source point p

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

G uu G Q dV u G dSn n

part part= minuspart part

part partrArr minus = minuspart part

int int

r r r r

r r r r r r r

r r r r r r r r r r r r

st implicit BC at infin

V2u Qnabla =

u gnpart

0 0 0 0 0

3D problem becomes 2D problem (BEM)

( ) ( ) ( ) ( ) [ ( ) (

for the 3D space will be shown next

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

minusint intr r r r r r r r r r r r

r rr r

source with -6 dBoct decay)

Physical interpretation monopole dipoleGGn

partpart

BCssource

( ) ( ) ---------(1)Due to spherical symmetry the solution has no angular dependenc

Free-spa

e ie( )

1 ( ) 0

ce Greens function (Laplace equati

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

By the divergence theorem

( ) -----------(2)

We will determine the constants that account for the singularity at the source by integrating Eq (1) around a small sphere

cG r cr

nabla =int

2 ˆ ( ) 1V V S

GdV G dV n GdSnabla = nabla nabla = nabla =int int int

0r ( )0rr minusδ

Spherical coordinates

2 2 11 120 0

ˆ (directional derivative at the radial direction)

1ˆ (4 ) 1 or lim -------------(3)4

From (2) and (3)1 1 lim lim ( )

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

part partnabla = =part part

part partnabla = = =part part

minuspart = = minus = rArr = minuspart

(point source)

1s ( )4

For simplicity we let 01

This free-space Greens function satisfies the radiation

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

Connection between Greenrsquos functions and eigenfunctions2

We consider the related eigenfunctions that satisfy the homogeneous problem ( )

eigenfunction expansio st homogeneous BCs

Apply the to the Greens function

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

φλ φ φ φ δ φλ φ φ

nabla = minus

nabla = nabla = = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

Thus ( ) bilinear expansion

1 symmetric ( is self-adj

( ) ( )( )

Reciprocitoint ) 2 cf EVD of symmetric matrices in linear

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Greens function for finite - boundary problems

Usally only one condition of and is given on the boundary since these

two are not independent and must satisfy compatibility condtion on

Xpartpartuu Sn

S1 ( ) ( ) (Integral eq of the 2nd kind)2

part part= minuspart partint

G uu u G dSn n

2 0unabla =

1 on Let

1 free - space GF (provide singularity) 4

homogeneous term due to boundary effect

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

st 0 on ( on ) By Greens 2nd indentity

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

part partnabla minus nabla = minuspart part

part partnabla +nabla minus nabla = minuspart part

int int

TT T T

T TV S

G S H G S

G uu G G u dV u G dSn n

G uu G H G u dV u G dSn n

u dV f ( ) partrArr =part partint intT T

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

hown ( ) (Rayleighs integ

part =part

part part partnabla = = = minuspart part part

u g Sn

HG H GH S Sn n

ng G dSr

For simple geometries eg planes spheres the homogeneous term can be found by using t method of imahe Omi

esttedrarr

Integral Transforms applied to PDE (infinite domain problems)

boundary discrete spectrum transf Fourier

FiniteSemi-infiniteInfinit

sinecosine transfIntegral transform

transf continuous spectru

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

rarr rarrrarrrarr rarr

Finite boundary problem

Semi-infinite problem infin

Infinite domain problem infinminusinfin

( )( )

1 transform domain1 ( ) ( ) ( ) ( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Ex Heat conduction in an rod diffusioninfinite hea

Fourier infin

xFi x i x

F f x e dx f x F e d

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

minusinfin minusinfin

rarrplusmninfin

larr⎯rarr

prime rarr

int int

quation ( ) temperature

( 0) ( ) (IC)

thermal conductivity mass density specific hea

st u x f x x

α minusinfin lt lt infin lt lt infin

= minusinfin

part part=part part

lt lt infin

Sol This is an infinite domain problem ( ) Requiring 0 or (also uniform convergence)

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

minusinfin lt ltinfinpart part rarr rarrplusmninfin

part part part= rArr = =part part part

part partQ

xu x u x

u u dF F i U t U t U tx t t dt

F t( )

2 2 st

changable parameter variable

0 (1 -order const coef ODE)

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( ) (look up math table)2

1 ( )2

convolution theoremα ω

ξ ξα π

minus minus minus minus

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

(Gauss distribution)

Diffusion is a smoothing process

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

Ex Heat conduction in

Unilateral Laplace transfoLaplace semi-infi

rmnite

st a iL sta i

n n n n

F s f t e dt f t F s e dsi

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

need 2 end conditions Sol or Use

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

part part= lt lt infin lt lt infinpart part

= lt lt infin= lt lt infin

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

partrArr = minuspart

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s A s e B s eu x t

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

rArr = sdot + sdot ( ) 0

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

On the other hand for another BC (0 ) ( ) (0 ) ( )

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

( ) (look up math table)2

HW Greenberg sec144 9(a) 11(a) Greenberg Appendix BC

xeh tt

x eh t d

τ ττα π

= lowast

= minus

Ex problem (Laplace transform) A string is initially at rest Its left end is fixed A force ( ) moves with

semi-infiniteconcentrated constant velocit

beginn

ng source

at 0 at ty

hF v v

e point 0 Find the vertical displacement ( )

( ) ( )

xy x t

xf x t F v x vt F v v tv

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = minus minus

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

δ ⎛ ⎞minus⎜ ⎟⎝ ⎠v

(wave speed in the string)

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

part part= minus =part part

= lt infin rarrinfinpart= =part

sform wrt time ( ) ( )

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

(time shifting property)

Complimentary solution ( ) ( ) ( )

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

Particular solution ( ) ( ) Substituting this into Eq(1) leads to

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

Y x s Y x s Y x s A s e B s e K s ey x t x Y x s x

minus minus

rArr =minus

= + = + +

lt infin rarrinfin ltinfin rarrinfinrArr equiv

( ) ( ) ( )s sx x

a vY x s A s e K s eminus minus

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

rArr = + = rArr = minus

= minus +

= minus

= minusminus

1) ( )

( ) ( )

where ( ) is the unit step function

u t es

v F x x x xL y x t t u t t u ta v v v a a

ττ τ

⎛ ⎞minus minus =⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

= minus minus minus minus minusminusuuur

2 When ( case p768Wylie) resonance

annihilator Laplace transform variation of param

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

by variation of parameters

BC (0 ) 0 ( ) 0

Thus ( ) 2

Note the discontinuity at

xY x s

F x eas

Y s A sFY x s x e v a

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Illustration of the results

Note case ( ) is acrossTransonic sound barrier

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Ex Concord jet 協和式客機

Moving source

It is easier to visualize the wave form by writing1 1 as (

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus minusminus minus minus minus

minus minus= minus minus = minus minus

x xt u t x vt u x vtv v v v

u x u x u x vt u x vtv v

1 (subsonic)v alt 2 (transonic)v a=

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

1 1x u xv vminus minus⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) ( )1 1x v t u x v tv vminus minus⎡ ⎤ ⎡ ⎤minus minus⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

3 (supersonic)v agtx xt u tv v

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) ( )1 1x a t u x a ta aminus minus⎡ ⎤ ⎡ ⎤minus minus minus⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

x x x xt u t t u tv v a a

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞minus minus minus minus minus⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ) ( ) 0tu t tu tminus equiv oF

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Non-homogeneous Problem nonzero equation andor BCrsquos

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HW (PDE Wylie)

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Connection between Greenrsquos functions and eigenfunctions

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Physical systems (with ldquodistributedrdquo parameters) depend on not only the ldquotime variablerdquo but also the ldquospace variablerdquorarr Governing equations are PDErsquos cf ODE for ldquolumprdquo parameters 1 independent variable t

Modeling governing equations see examples in Sec 112

Define problem apply physical laws PDE solution

Vibrating string

( )( )

y x tT

( ) f x y y tamp

T tension force

sin sin | tan |

Newtons law F

x x x x

m xf x y y t

T T TT T T

α α αα α α

+Δ +Δ⎧⎨⎩

Δ = Δ

uarr + =

partΔ

tan | tan | ( )

tan | tan |( ) ( )

( 0) (tan ) ( )

( ) ( )

y T T f x y y t xt

y T f x y y tt x

x T f x y y tx

α αρ

= minus + Δpart

( ) f x y y tamp

y T y f x y y

fρ ρ

part part

part partminuspart

=part part

2 2 2 22 2 2

2 2 2 2

2 2 2 2 22 2 2

2 2 2 2 2

ve equation ( )

t x y tp p x y z t

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

---------(1)

homogeneou

t ( ) ( )

y x t y

c ct t t

ζ ηζ η

ζ η ζ ηζ η

ζ η ζ η

ζ η= minus

part part

2 22 22 2

y yc c y c c yx t

ζ η ζ η⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2ζpartpart

ηζ ηpartpart

part+ +part part

2y c⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2ζpartpart

ηζ ηpartpart

y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

= minus +

rArr = + = +

rArr +int

Consider an string

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

part= =part

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Vibrating string

( )( )

y x tT

( ) f x y y tamp

T tension force

sin sin | tan |

Newtons law F

x x x x

m xf x y y t

T T TT T T

α α αα α α

+Δ +Δ⎧⎨⎩

Δ = Δ

uarr + =

partΔ

tan | tan | ( )

tan | tan |( ) ( )

( 0) (tan ) ( )

( ) ( )

y T T f x y y t xt

y T f x y y tt x

x T f x y y tx

α αρ

= minus + Δpart

( ) f x y y tamp

y T y f x y y

fρ ρ

part part

part partminuspart

=part part

2 2 2 22 2 2

2 2 2 2

2 2 2 2 22 2 2

2 2 2 2 2

ve equation ( )

t x y tp p x y z t

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

---------(1)

homogeneou

t ( ) ( )

y x t y

c ct t t

ζ ηζ η

ζ η ζ ηζ η

ζ η ζ η

ζ η= minus

part part

2 22 22 2

y yc c y c c yx t

ζ η ζ η⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2ζpartpart

ηζ ηpartpart

part+ +part part

2y c⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2ζpartpart

ηζ ηpartpart

y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

= minus +

rArr = + = +

rArr +int

Consider an string

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

part= =part

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

投影片編號 49

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sin sin | tan |

Newtons law F

x x x x

m xf x y y t

T T TT T T

α α αα α α

+Δ +Δ⎧⎨⎩

Δ = Δ

uarr + =

partΔ

tan | tan | ( )

tan | tan |( ) ( )

( 0) (tan ) ( )

( ) ( )

y T T f x y y t xt

y T f x y y tt x

x T f x y y tx

α αρ

= minus + Δpart

( ) f x y y tamp

y T y f x y y

fρ ρ

part part

part partminuspart

=part part

2 2 2 22 2 2

2 2 2 2

2 2 2 2 22 2 2

2 2 2 2 2

ve equation ( )

t x y tp p x y z t

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

---------(1)

homogeneou

t ( ) ( )

y x t y

c ct t t

ζ ηζ η

ζ η ζ ηζ η

ζ η ζ η

ζ η= minus

part part

2 22 22 2

y yc c y c c yx t

ζ η ζ η⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2ζpartpart

ηζ ηpartpart

part+ +part part

2y c⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2ζpartpart

ηζ ηpartpart

y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

= minus +

rArr = + = +

rArr +int

Consider an string

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

part= =part

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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y T y f x y y

fρ ρ

part part

part partminuspart

=part part

2 2 2 22 2 2

2 2 2 2

2 2 2 2 22 2 2

2 2 2 2 2

ve equation ( )

t x y tp p x y z t

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

---------(1)

homogeneou

t ( ) ( )

y x t y

c ct t t

ζ ηζ η

ζ η ζ ηζ η

ζ η ζ η

ζ η= minus

part part

2 22 22 2

y yc c y c c yx t

ζ η ζ η⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2ζpartpart

ηζ ηpartpart

part+ +part part

2y c⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2ζpartpart

ηζ ηpartpart

y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

= minus +

rArr = + = +

rArr +int

Consider an string

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

part= =part

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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---------(1)

homogeneou

t ( ) ( )

y x t y

c ct t t

ζ ηζ η

ζ η ζ ηζ η

ζ η ζ η

ζ η= minus

part part

2 22 22 2

y yc c y c c yx t

ζ η ζ η⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2ζpartpart

ηζ ηpartpart

part+ +part part

2y c⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

2ζpartpart

ηζ ηpartpart

y⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

= minus +

rArr = + = +

rArr +int

Consider an string

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

part= =part

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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24 0 ( )

( ) ( ) ( ) ( ) ( )

---------(( ) ( ) ( ) 2)y x t F x ct G x c

y f d G

ζζ η ζ

= minus +

rArr = + = +

rArr +int

Consider an string

( ) ( ) ( ) ( ) (

infini

0) ( ) ( )

y yc x tx t

yy x p x x q xt

part= =part

(3)( ) ( ) ( ) ( ) ( ) ( 0)

( ) ( )

( )( ) ( )( ) ( ) ( ) (4)t

(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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(4) ( ) ( ) (0) ( ) (0)

( ) ( ) (0) (0) ( ) (5)

(3)(5) ( ) ( ) ( )

( ) ( ) (0) (0) ( )

( ) 1 ( ) (2 2

F x G x p x

F x G x F G q x dx

p xF x q xc

primerArr = minus

intint

(0) (0)) (6)2

( ) 1 (0) (0) ( ) ( ) (7)2 2 2

(2)(6)(7) ( ) ( ) ( )( ) 1 (0) (0) ( )

p x F GG x q x dxc

cminus

= minus + +

( ) 1 (0) (0) ( )2 2 2

Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Thus ( ) ( ) 1( ) ( )2 2

( ) ( ) ( )2

minus + +=P

( )p x

are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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are constants

( 2 ) ( ) 0 2 0 for nontrivi

u u uA B Cx x y y

part part part+ +

Δ =2 2

hyperbolic

parabolic

if liptic 0 eg

ACu uc

x tu ua

x tu u

1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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1 1 2 2

1 1 2 22

1 1( ) 2 2

i ii i

C B Ax y c x y c

A y By C A B C

λ λ λ λλ λ

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+ + =+ = + =

( 2 ) 0 12i i

A y By C

A B C iλ λλ

+ += = =

( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) 2 ( ) ( ) ( )

dent sol

ions of ( ) )

A B C x

tions of

( )( ) 2 ( ) ( ) 0 Then

a ( ) ( ) ( )

(most useful

hyperbolic

b ( ) ( )

( ) ( ) (

) ( )c 2 2

x y x y x y x yr si

φ ψ φ ψ

+ minus= =

hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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hyperbo0 0 PDE2 4

( ) 2 0 ( ) 0 ( ) 0

u u ux yx x y x

y yA x B

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎧⎨⎩

prime =rArr rArr

prime+

ln ln ln0

characteristic( )

Hyperbolic PDE Let

⎧⎧⎪ ⎪

⎨ ⎨⎪⎩⎪

⎩⎧⎨⎩

= =rArr

+ =+ =

there4 = =

u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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u u rx

ypart part

part=part

part part2 2

u s uys x s

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2 2 2 2 2

u s uys x s

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠ ⎣ ⎦

⎡ ⎤⎢ ⎥⎣ ⎦

partpart

uyspartminuspart

2u uy xy

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

part+part

0 or 0

( ) ( ) ( ) ( ) ( ) ( )

u uyr s r s

rArr = +rArr = +

ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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ibrating shaftn

a x lt xEa

( 0) ( ) and ( 0) (ICs )

x f x x g xt

ρθθ part= =part

Fixed-free

= =part part

= =part

0 lBCs

( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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( ) ( ) ( ) ( )( ) ( ) separation con ( )( ) ( )

indep of

( ) ( )

and ( ) ( )

x t X x T t

T t T t X x X xa

θθ θ

rArr = =

physically impo

t t x a x a

x a x a t t

T T X Xa

λ λ λ λ

μ λλ

plusmn

minus minus

rArr = + = += + +

Q 對稱

( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) ( ) ( ) ( )

( ) cos

T t T t X x X xa

T t A t B t X x

μλλ

= minus gt

) cos sin

( ) ( ) ( ) cos sin ( cos sin )

C x D xa a

λ λθ λ λ

πλλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = + +

rarr = =

fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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fos sin ) 0

If 0 then ( ) 0 Thu

ixed-fixe

( ) ( in )(

x t D xa

λθ θ

θ λ λθ

rarr rarr= =

rArr = cos sin )

BC2 ( ) 0 ( sin )( cos sin )

If 0 then ( ) 0 If 0 then ( ) 0

A t B t

l t D l A t B ta

lll n n

λθ λ λ

λ λ λπ

equiv = +

equiv = =

sin ) ( ) ( ) ( cos sin )

nn n n n n n nx

λ πλ λ λ⎛ ⎞⎜ ⎟⎝ ⎠

= = + =

(eigenvalues)

( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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( 0) ( ) and ( 0) ( )

x f x x g

θθθ

θinfin

part= =part

rary That is

( ) ( ) sin ( cos sin )

( 0) ( ) si

Fourier

n n nn n

n xx f x Al

π π πθ θ

infin infin

sum sum

l lπ= int 章節

Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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Fourier si

( ) sin sin cos

( 0) ( ) sin

ne series

n a n xx g x Bt l l

g x ln a B

θ π π π π

θ π π

=infin

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

2 ( )sin

2or ( )sin

) sin ( cos sin )

2 2( )s

Note Sim

in ( )sin

l ln n

n x n xA f x dx

n xg x dxl l

n xB g x dxn a

B g x dxl l n a l

π π πθ

π ππ

θinfin

BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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BCs ICs expansion n

⎯⎯⎯⎯⎯⎯⎯rarr

⎧⎨⎩

rarr rarr

x n xa l

⎫⎪⎪⎬⎪⎪⎭

orthogona

tions Be

rarr Q

Legendre Chebyshev

= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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= =rarr

If 0 applying (0) 0

A x B xy xC Dx

BC sy C

C D y xy l C Dl

λ λ λλ

⎧⎪⎨⎪⎩

⎫⎬⎭

+ ne=+ =

If 0 applying (0) 0

BC sy A

B ly l A l B lB y x

λλ λ

⎫⎪⎬⎪⎭

( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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( ) sin ie ( ) sin

l l n n n nn

ln x n xy x B x

l lm n

x x dx l m n

λ λ ππλ

π πφ

⎧⎪⎨⎪⎩

= rArr = = = lt

rArr =

ne= rArr

xEigenvalue pro

ndentaln

λ λλ λ

⎧⎨⎩

rarrΔ

Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Greenberg p497

( ) 0 ( ) 0 on [ ] are o

p x r x a b

λα β γ δ

primelt infingt gt

(function space)

cf in the vec

( ) ( )

r produ

au v u x v x r x

sdot sum

intDef

Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Norm ( ) ( )

( ) ( ) ( )

u u u u x r x dx

α α α

= hArr

intint

eigenva

tten a

lue prob

d dL p qr dx dx

⎡ ⎤⎛ ⎞⎢ ⎥⎜

⎤⎣

⎟⎝ ⎠ ⎦

primeprime + + =

minus +

Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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Lu v u LvL SLP

pf 1 = ( ) [( ) ]

ru v BCs x a b

int int

( ) ( )

[ ( )] [( ) ]

p uv u v pv qv u dx

( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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( ) ( ) 0 ( ) ( ) 0If 0

( ) ( ) ( ) ( )

If 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0

Hence ( ) 0

x b x b

u v u v

BC su b u b v b v b

u b u b v b v b

uv u v

y b u b v b uv u v

uv u v

δ δγ γ

γ δ γ δγ

δ δγ γ

larly ( ) 01Thus 0 [( ) ] [( ) ]

uv u v

u Lv r dx u Lv

lowast

int int

(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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(1) (2) ( )

eigenvalues r

φ λφ

minus minus = minus

0 ie QED

λ λ λ

there4 minus = isin

orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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orthogEig

j j j k k k

j j j j jk k k

= = (1)

(2) (1) ( )

0 ( Thm1)

0 ( )j

j j jk k k k k

j j j jk k k k

j jk k

λ λ φ φ

λ λ λ λ

φ φ φ

minus = minus

= minus

minus = minus ne

primethere4 =

are orthogonal QED

piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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piecewise continuou

(lete Wylie) and

Pf Let ( ) ( )

f x c x

f c c c

φ φ φ φ φ φ

φφ φ

=infin infin

+ minus infin

⎡ ⎤⎣ ⎦

sum Q( )

rthogo

ntinui

nality

φ φφ φ φ

rArr = =

ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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ial fun

alitm ho

lowast

( ) sin for 123

py qy ry py

λπ πλ φ

rArr = = =

p q r a b l

λα β γ δ

α γ β δ

= = = = = = = = = rarr

0 ( ) ( ) sin sin (Thm2)

ne= = =

=int int

b lm n m na

sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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sin ( 1)

( 1)( ) ( ) sin

nn nn n

f x xn x lf x dx

l nn x dx

ll nf n xf x x

πφπ

πφ φ

πφ πφφ φ

+infin infin

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= = minus

minus= =

sum sum1

2 ( 1) sin

l n xn l

+infin

minus=

( )f x( )f x x=

( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) ()

(0 ) 0 (0 ) 0

part= = =part

比前例

u ua F x tt x

uu t M EI tx

2 ) 0 ( ) 0

IC ( 0) ( )

( 0) ( )

part= = =part

=part =part

ut M EI l tx

u x f xu x g xt

moment

0x= x l=u

Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Sol ( ) ( ) ( )

homo PDE 0

01 =gt

=gt 0 ( ) cos sin

φ φφ βφ

β ωω ω

+ =minus= =

+ + == +

ampamp

ampamp ampamp

代入

u x t x g tu ua

t xg a g

(0 ) (0) ( ) 0 (0 ) (0) ( ) 0 BC

(must be homo

( ) ( ) ( ) 0

minus = = + + +

xuu l t l g t

x( ) ( ) ( ) 0

or (0) 0 (0) 0 ( ) 0 ( ) 0

φ φφ φ

⎧⎪⎪⎨⎪⎪⎩

primeprime= =

l t l g t

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

投影片編號 49

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投影片編號 79

(4) (3)4 4

pf ( )

d x Ldx

β φ φ

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

minus =

BC (0) 0 (1) (0) 0 cos

dx u v uv dx

d vu Lv u v dxdx

Lu v u Lv L x x

A CA x C

λ φ φ

φ β β β

⎛ ⎞⎜ ⎟⎝ ⎠

= minus

= rArr isin perp

= = + minusminus

int int

代入

cosh ( ) (2)x

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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投影片編號 79

sin 0 12

2 sin( )

β β ππβ ω β

⎧⎪⎪⎪⎨⎪⎪⎪⎩

= rArr = =

rArr = =

B l l n nn al

n xl l

(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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(4) 4 440 0

(modal analysis

φ φ φ φ δ

=infin

int int

sum ampamp代入

lm n m n mn

d dx dxdx

u x t x g t

1 12 4

decoupled (ODE)

φ φ φ φ φ

φ φ φ φ

infin infinlt gt

infin infin

rarr + =lt gt

+ = rArr

sum sum

ampamp

m n n m n n m mn n

m n n m n n mn n

m m m m

g F x t

g a g F Q

g a g Q

mode shapes

modal coordinates

IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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IC ( 0) (0) ( )

(0) (0)

( 0) (0) ( )

(0) (0)

Solve ( ) ( ) ( ) 12

φ φ φ

rArr = =

part = =part

rArr = =

amp amp

ampamp

m m n n mn

u x g f x

u x g g xt

g t g t Q t m

φinfin

⎫⎪⎬⎪⎭

g fg t

u x t x g t

2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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2 21 2

0 st 0 st

u Q uu u

u uu Q ux

u u u PDE Cu

nabla = nabla == =

rarrnabla =

1 2 3 4

(0 ) ( ) nonseparable ( ) ( ) ( 0) ( ) ( ) ( )

u ux y

u u u u u

⎫⎪⎪⎬⎪⎪⎭

1u g=2u g=

2 0unabla =

2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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2 24 4

st (0 ) ( ) ( ) 0 ( 0) 0 ( ) 0

= == =

( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) 0 (0) 0 ( ) 0 ( ) ( ) ( ) ( ) 0( ) ( )( ) ( )

( ) ( ) 0( ) ( ) 0

In the -directio

φ φφ φ

φ λφλ

φ λφ⎧⎨⎩

== = =

primeprime + =

h L Hy h x h x y

h x yh x y

h x h xy y

st (0) 0 ( ) 0

( ) sin 123

φ λφφ φ

⎧⎪⎪⎨

⎛ ⎞⎪ ⎜ ⎟⎪ ⎝ ⎠⎩

primeprime + == =

n yy nH

1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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1 2 1 2

( ) cosh sinhsinh 2

sinh( ) cosh sinh

λπ π

ππ π

minus⎛ ⎞⎜ ⎟⎝ ⎠

rArr = minus +

Let ( ) sinhcosh

sinhcosh

n n nn n

a na h x a x Ln HLH

a n x Ln HLH

π πφ

infin infin

= = minus

= minus

sum sum

BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

投影片編號 49

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BC (0 ) ( )

sin sinh ( )

2 ( )sinsinh

n yA g y dyn L HHH

u u u u u

rArr minus =

minusrArr =

rArr =

1 2 3 4u u u+ + +

nabla ==

= nabla = ==

u u Q uu

Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Let ( ) ( )sin

sin sin

π π π

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

rArr minus

d b n n xbdy L L

( ) ( ) (

π π⎛ ⎞⎜ ⎟⎝ ⎠

minus minus= +

minus =

int int

intn n

n n ny

u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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u Q ux y

x y x y x y

n mL H

φφ λφ φ

π πλ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nabla = =

nabla = minus =

rArr = +

1 1 1 1

( ) sin sin ( )

Let ( ) ( ) sin sin

nfunction expansion

mn mn mnn m n m

n x n yx yL H

= = = =

= =sumsum sumsum

( ) ( )1 1 1

infin infin infin

sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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sin( )sin( )

sin ( )sin ( )

sin( )sin( )

( ) sin s

λ π π

λπinfin infin

minus =

rArr =minus

=minus

int int

sumsum

mnmn mn

mn mnH L

mn H L

n x L n y H dx dy

Q n x L n y H dx dy

LHn xu x y bL

πn yH

HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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HW (PDE Wylie)

11-3 3 1511-4 1 4(d)11-5 4 6 11(b) 3111-6 4 5 40

sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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sion ction method

non-separable

IC ( 0) ( )

( 0) ( )

u uc Q x tt x

u t A t u L t B t

u x f xu x g xt

=part =part

lowast

eens identity

2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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2 2(free vibration)

Homogeneous Problem

Eigenval

0 key pt homo prob

st (0 ) ( ) 0Let ( ) ( ) ( )

0( ) 1 (( ( )

nu uct x

x b tx c b t

φφ φφ λφ

ampamp

只有才找的到

( ) ( ) 0

st (0) ( ) 0 ( (0) ( ) ( ) (

)n n m n mn

d x xdx

L b t L b t

n n x LxL L

φ λ φ

φ φ φ φ

π πλ φ φ φ δ⎛ ⎞⎜ ⎟⎝ ⎠

= = = equiv

rArr = = rArr =

49(Natural modes)

2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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2 2 21

2 22 2

2 22 0

1 modal coo( )

( ) ( ) ( )

( )( )

( ) ( )

u uc Q x c Qx x

u x t b t

b tdt x x

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

part +part

uuuuuuur 20

( ) ( ) ( )

( )( )

( ) ( ) ( )

n n Ln n n

Q x t x dxQq t

Q x t q t x

φφφ

partpart= +

intint

uuuuuuur

x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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x L xu x t

u x t x u x t x

2 2 21 1

homoge

nn n n

φφinfin infin

以方波為反例

Greens 2 identits)

LL Ln n n

u u udx dxx x x x

uu dx u dxx x x x

φφ φ

φ φ φ

int int

int int51

2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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2 20 00

LL L n n

Ln n n

LL n n

u udx u dx ux x x x

uu dx ux x x x

φ φφ φ

φ φ φ

⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= minus

part int intint

( ) ( )

Lu dxφ

φλ φinfin

sumint ( ) ( ) ( ) (0)n

nu L t u L t Lx x

φ φ⎡ ⎤⎢ ⎥⎣ ⎦

0 ( 1)

( ) sin

( ) co

) ( ) cos

) (0 ) (0)

( ) co

Ln n n

u t u tx x

n nn n xB tn xxL

n nb t dx A t B t n

d b q tdt

L LLπφ

π πλ φ

ππ π

= minus

⎧ ⎫⎨ ⎬⎩

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜

⎟⎝⎝

⎟⎜⎠⎠

= minus + minus

( ) 2( ) ( ) ( ) ( )( 1)( )

nn n nL

φ πλφ

⎡ ⎤⎣ ⎦

intint

IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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IC (modal space) ( 0) ( ) (0) ( )

(1)nnn n n

source BC

u x f x b x

φinfin

= =sum

144424443

( ) ( ) (0)

( ) ( )( 0) ( ) (0) ( ) (0)

n n n Ln n

f x x dxb

there4 =

intint

intsumint

qs (2) and (3)

u x t b t xφinfin

Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Define the

nonhomogeneous

free-spa Gce

impulse

reens fu

Greens funct

nction ( ) ( ) c

uu Q u f g Sn

partnabla = = =part

nabla = minus

r r r r

2 20 0 0 0 0

response

y[ ( ) ( ) (

) ( )] ( )

[ ( ) (

u G G u dV

nabla minus nabla

= minus

r r r r r r r

r r r 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

) ( ) ( )] ( )

[ ( ) ( ) ( ) ( )] ( )

( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )] ( )

G Q dV

G uu G dSn n

int int

r r r r

r r r r r r r

V2u Qnabla =

u gnpart

0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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0 0 0 0 0

( ) ( ) ( ) ( ) [ ( ) (

(Free-space GF a po

) ( ) ( )]

Gu G Q dV f G g dSn

πminus

part= +

r rr r

partpart

BCssource

Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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Free-spa

e ie( )

1 ( ) 0

G G rd dG rr r

r dr dr

⎛ ⎞⎜ ⎟⎝ ⎠

nabla = minus

= = minus ne

r r r r

( ) -----------(2)

cG r cr

nabla =int

2 ˆ ( ) 1V V S

0r ( )0rr minusδ

2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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2 2 11 120 0

1ˆ (4 ) 1 or lim -------------(3)4

G Gn Gn r

G Gn GdS r rr r

cGr r c cr r

rarr rarr

(point source)

1s ( )4

1 ( )4

dition at infinit

Sommerfel

G r cr

rarrinfin

minus=

0⎞⎜ ⎟

φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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φ λφ

nabla =

nabla = r

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

n n n n n n nn n n

m n n n mn

mm m m m m m m

m m mV

Ga a a

a dV a

δφ φ λ φ δ

φ λ φ φ δ

nabla = minus

rarr = minus

= minus = rArr =

sum sum sum

r r r rr r r r r

rr r r r r

( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) ( )( )

( )alg

( )ebra

n nn n

φ φφ

nabla=

sumsum0

r rr r

Xpartpartuu Sn

G uu u G dSn n

2 0unabla =

1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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1 on Let

Dirichilet problem

satisfies 0 (homo)

minus=

nabla =

u f SG G H

( )2 2

0 0 02 2 2

= = = = =

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤ ⎛ ⎞⎢ ⎥ ⎜ ⎟

⎝ ⎠⎣ ⎦

= = minus

int int

TT T T

T TV S

G S H G S

G GdS u f dSn n

Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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Choose to satisfy

0 st 0 on ( on )

In this

Neumann probl

it can be s

part =part

u g Sn

HG H GH S Sn n

ng G dSr

esttedrarr

Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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Fourier seri

esLaplaceFourie mr

n nλ φ⎫⎬⎭

( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( )( )

2 ( ) ( )

if lim ( ) ( ) ( ) 0

Fourier infin

xFi x i x

F f x i F

f x f x f x

ω ωω ω ωπ

infin infinminus

rarrplusmninfin

larr⎯rarr

prime rarr

int int

( 0) ( ) (IC)

st u x f x x

= minusinfin

part part=part part

lt lt infin

( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) ( ) ( )

and are inter

α α ω ω ω ω⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part partQ

xu x u x

F t( )

2 2 st

( ) ( )

IC ( 0) ( ) ( ) ( 0) ( ) ( ) ( )

( ) ( )

ωω ωω ω

rArr + =

rArr =

= =rArr =

rArr =

td U UdtU t A e

F u x F f x FU F

U t F e

( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) ( ) ( )

1 ( )2

ξ ξα π

infin minus minus

minusinfindarr

⎯⎯⎯rarr

⎛ ⎞⎜ ⎟⎝ ⎠

= lowast

= int 14243

xF tx x

u x t F F F e

f x et

f e dt

( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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( ) 1 ( 1)

( ) ( )

2 transform domain

1( ) ( )2

( ) ( ) (0) (0)

rmnite

st a iL sta i

n n n n

L f t s F s s f f

minus larr⎯rarr

= minus minus minus

= intint

14444244443L

(0 ) ( ) 0 (BC) ( 0) 0 0 (IC)

d 1 end condition

xx t t

u u x tx t

u t h t tu x x

LL L L

⎫⎬⎭

( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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HW (PDE Wylie)

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( ) ( ) ( 0)

t tu uL L

U x s sU x s u xx

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

part part=part part

( ) ( ) ( )Impose lim ( ) 0

or lim ( ) 0 0

0 ( ) lim ( ) l

icit B

s sx x

d U sUdx

U x s L

A s e B s e

rarrinfin

rarrinfin rarrinfin

rArr minus =

rArr = +=

Hence ( ) ( )s x

U x s B s e αminus

rArr =

convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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convolution t

(0 ) ( ) ( )

( ) ( )Thus

( ) ( ) ( heore )m

L u t L h tU s H s

U s B s H s

U x s H s e

u x t h t L e

minus minus

rArr = =

rArr = sdot

= lowast2 2

xeh tt

x eh t d

τ ττα π

= lowast

= minus

beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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beginn

ng source

at 0 at ty

hF v v

( ) ( )

xy x t

xF v v t F vv

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

= minus = 1v

δ ⎛ ⎞⎜ ⎟⎝ ⎠minus

0xF tv

st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

投影片編號 2

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HW (PDE Wylie)

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st (0 ) 0 ( ) as (BC)

( 0) 0 ( 0) 0 (IC)

Taking Laplace tran

1f x t

y y Ta at x

y t y x t x

ρ ρ⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( 0)

tt Y x s L y x t

s Y x s s y x

minus ( 0)y xt

partminuspart

s sx xa a

Fd Ya x s edx

Fd Y s Y edx a a

Y x s A s e B s e

= minus

rArr minus =

rArr = +

1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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1 If ( or cases)

subsonic supersoni

Y x s K s e

s K s ev

v aminus

2 ( )s x

vs K s ea

minus 02

s xvF e

( )( )

Thus ( ) ( ) ( ) ( ) ( ) ( )BCs ( ) as or ( ) as ( )

implicit BC ( ) 0

s s sx x xa a v

v FK sa v s

v FY x s ea v s

minus minus

rArr =minus

= + = + +

( ) ( ) ( )s sx x

rArr = +71

(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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(0 ) ( ) ( )

(0 ) 0 or (0 ) 0

( ) ( ) 0 ( ) ( )

( ) ( ) ( )

s sx xa v

sx v sx a

s sx xv a

Y s A s e K s e

y t Y s

A s K s A s K s

Y x s K s e K s e

K s e e

v F e ea v sρ

minus minus

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= minus +

= minus

= minusminus

1) ( )

( ) ( )

u t es

ττ τ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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ansonic

s sx xa a

Fd Y s Y edx a

Y x s A s e B s e

minus =

lim ( )

BC (0 ) 0 ( ) 0

Thus ( ) 2

xY x s

F x eas

asF xy x t x u t

a ax at

rarrinfinltinfin

⎛ ⎞⎜ ⎟⎝ ⎠

minus sdot

= rArr =

rArr = minus sdot =

= minus sdot minus

Q64748

1442443

( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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( )3 subsonic4

( ) transonicv a=

( )5 supersonic4

discontinuous

Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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Shock th

wavdis

e contin

N-wavee

Sonic boom

sy x t

Cone of shock

Moving source

1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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1 1Also note th

ow to p

( ) ( ) ( ( ))

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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x1 xvminus

vminus

1u xvminus⎛ ⎞

⎜ ⎟⎝ ⎠

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1vminus

xxu ta

⎛ ⎞minus⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞minus minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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PARTIAL DIFFERENTIAL EQUATIONS (PDE)ocw.nctu.edu.tw/course/engineeringmathematicsII/ch06.pdf · partial differential equations (pde) 機械工程學系 白明憲 教授

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PARTIAL DIFFERENTIAL EQUATIONS (PDE)ocw.nctu.edu.tw/course/engineeringmathematicsII/ch06.pdf · partial differential equations (pde) 機械工程學系白明憲教授