08 Wave Equations

8/20/2019 08 Wave Equations

1/13

Wave Equations


2/13

UCF Wave Equations (I)

likewise


3/13

UCF Wave Equations (II)


4/13

UCF Scalar Wave Equation

equationescalar wav1D 01

havewe,0and 0oronly,zondependsthatAssume

equation.wave3DaisThis.1

where

01

havewe,ororeitherrepresentLet

equation.escalar wavsatisfyshould field electricof componenteachThen

s,coordinaterrectangulaIn

0:equationector waveConsider v

2

2

22

2

2

2

2222

2

22

t u

v zu

y

u

x

uu

v

uv

u

u

t

p

p

p

z y x

z z y y x

EEE

EEE aaax

E

E E

z

0 ,0

y x


5/13

UCF General Solution to 1D Scalar Wave Equation

solution.aalsois)/( provealsocanweLikewise,

,11

,

1

.1and 1

havewe,Let

:solutionais)(Prove

).()(),(:issolutiongeneralthe

01

:equationescalar wav1DFor

2

2

2

2

2

2

22

2

2

2

22

2

p

p p

p

p p

p

p p

p

v zt g

u

t

u

t

uu

v z

u

v z

u

u

t

u

t

uu

v z

u

z

u

t v zv

zt

v

zt f

v zt g

v zt f t zu

t

u

v z

u


6/13

UCF Physical Understanding of General Solution

)(Plot pv

zt f

z

)( pv

z f

)(

)(

p

p

p

vt v z f

v

zt f

t v z p10 .itywith velocdirectioning propagatinwaveaforstands :)( p

p

v zv

zt f

“backward wave”, or “negative wave”, or “reflected wave”

.itywith velocdirectioning propagatinwaveaforstands :)( p p

v zv

zt g

“forward wave”, or “positive wave”, or “incident wave”Likewise:

lightof yvelocitm/s10997.2 111 8

00 c

cv

r r r r

p


7/13

UCF Sinusoidal Case

)cos()cos(

])(cos[])(cos[

),(),(),(

B A

B

p

A

p

kzt Bkzt A

v zt B

v zt A

t zut zut zu

frequency.spatialorrwavenumbeisfrequency,angulariswhere pv

k

In general, we can use plane waves to expand any arbitrary wave which

is the Fourier Transform in both time and spatial domain.


8/13

UCF Period and Wavelength

A

-A

)cos(),(Plot kzt At zu

k k T T T

2

,2 :lengthwave

2

,2 : period

z0


9/13

UCF Electromagnetic Spectrum


10/13

UCF Traveling Wave and Standing Wave

)cos()cos(

),(),(),(

B A kzt Bkzt A

t zut zut zu

wave.travelingais),(wavetotalthe

0,isorif Orwave.travelingais),(or),(Each

t zu

B At zut zu

wave.standingais),(wavetotal,If t zu B A

wave.standing-travelingacalled is),(Otherwise, t zu


11/13

UCF Standing Wave (1)

A2

2

A2

0 2/ 2/

When , 0 and 0, ( , ) [cos( ) cos( )]

2Asin( )sin( )

A B B u z t A t kz t kz

t kz


12/13

UCF Standing Wave (2)

z

))sin(Asin(2),(For kzt t zu

0 2/ 2/


13/13

UCF

Phasor Form Representation

)Re()Re

)cos()cos(

),(),(),(

t j jkz jt j jkz j

B A

ee Beee(Ae

kzt Bkzt A

t zut zut zu

B A

jkz jkz j

jkz jkz j

eU e BeU

eU e AeU

B

A

0

0

(propagating in +z direction)

(propagating in -z direction)

z

0 ,0

y x

08 Wave Equations

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