11/04/2013PHY 711 Fall 2013 -- Lecture 261 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 26: Traveling.

PHY 711 Fall 2013 -- Lecture 26 111/04/2013

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 26:Traveling and standing waves in 1 and 2 dimensions

1. Brief review of 1-d from Chap. 72. Waves in 2-d elastic membranes –

Chap. 8

PHY 711 Fall 2013 -- Lecture 26 211/04/2013

PHY 711 Fall 2013 -- Lecture 26 311/04/2013

equation. wave thesatisfies )()(

:or function any for that Note

0 2

22

2

2

ctxgctxfμ(x,t)g(q)f(q)

x c

t

Review of wave equation in one-dimension – here (x,t) can describe either a longitudinal or transverse wave.

Traveling wave solutions --

PHY 711 Fall 2013 -- Lecture 26 411/04/2013

ctx

ctx

x

x

x

dxxc

ctxctxtx

dxxc

xxg

dxxc

xxf

g(x)f(x)x

dxxc

xgxf

dxdg(x)

dxdf(x)cx)(x,

tμ

xgxfx)μ(x,

x)(x,tμx)μ(x,

')'(21)()(

21),(

')'(1)(21)(

')'(1)(21)(

: and find ,each For

')'(1)()(

)(0

)()()(0 : then

)(0 and )(0 :problem valueInitial

PHY 711 Fall 2013 -- Lecture 26 511/04/2013

Standing wave solutions of wave equation:

.00with

0 2

22

2

2

μ(L,t),t)μ(x

c t

c kρ(x)k

dxρ(x)d

ρ(x)etx ti

0 where

),( :Assume

22

2

ckL

k

LxAx

sin)(

PHY 711 Fall 2013 -- Lecture 26 611/04/2013

PHY 711 Fall 2013 -- Lecture 26 711/04/2013

Extension of ideas to wave motion on a two-dimensional surface – elastic membrane (transverse wave; linear regime).

u(x,y,t)

PHY 711 Fall 2013 -- Lecture 26 811/04/2013

0///

0

:principle sHamilton'

,,;,,,

,,;,,, :density Lagrangian

2

1

yuyxuxtutu

Ldtδ

dxdytyxtu

yu

xuuL

tyxtu

yu

xuu

t

t

LLLL

L

L

PHY 711 Fall 2013 -- Lecture 26 911/04/2013

2222

2

22

where 0

0///

21

21,,;,,,

: and constant with membrane elasticfor density Lagrangian

cuctu

yuyxuxtutu

ututyx

tu

yu

xuu

LLLL

L

ckyxk

yxetyxu

cuctu

ti

where 0),(

),(),,(

:solutions waveStanding

where 0

:equation waveldimensiona-Two

22

2222

2

PHY 711 Fall 2013 -- Lecture 26 1011/04/2013

Consider a rectangular boundary:

a

b

mnmnmn

mn

ckωb

na

mk

byn

axmAyx

bxxyay

sinsin),(

0),()0,(),(),0(:conditionsboundary Clamped

222

PHY 711 Fall 2013 -- Lecture 26 1111/04/2013

22

211

bak

2 222

21

bak 22 22

222

bak

2 222

12

bak

PHY 711 Fall 2013 -- Lecture 26 1211/04/2013

More general boundary conditions:

side free"" represents 0

spring with dconstraine side boundard represents

b

bb

u

uu

mnmnmn

mn

ckωb

na

mk

byn

axmAyx

xya

xybxx

sincos),(

0),(),0(),()0,(

:conditionsboundary Mixed

222

222

11

bak

PHY 711 Fall 2013 -- Lecture 26 1311/04/2013

Consider a circular boundary:R

0),(:),(for conditionsboundary Clamped

Rr

c

krk where 0),(22

integer )2()( :Note

)( :Let

)()(),( :Assume

11

system coordinate lcylindricaIn

2

2

22

22

m

e

rfrrrrr

im

PHY 711 Fall 2013 -- Lecture 26 1411/04/2013

)(function Neumann lCylindrica )(function Bessel lCylindrica

solutions ential transcendorder withinteger ofequation Bessel

0)(1

:function radialfor equation alDifferenti

22

2

2

2

zNzJ

rfkrm

drd

rdrd

m

m

Consider circular boundary -- continued

PHY 711 Fall 2013 -- Lecture 26 1511/04/2013

Some properties of Bessel functions

11.620,... 8.417, ,136.5 :210.173,... 7.016, ,832.3 :18.654,... ,520.5 ,406.2 :0

0 functions esselB of Zeros

2

1

0

n

n

n

mnm

zmzmzm

zJ

42cos2)( :form Asymptotic

)(2)()(

)(2)()( :relationsRecursion

2!!1

2)( :series Asending

1

11

11

2

0

mzz

zJ

dzzdJzJzJ

zJzmzJzJ

zmjj

zzJ

zm

mmm

mmm

j

j

jm

m

PHY 711 Fall 2013 -- Lecture 26 1611/04/2013

Some properties of Bessel functions -- continued

'2

1

2

0

'

2

: functions for theidentity

following theprove topossible isIt :Note

nnmnm

Rmn

mmn

m

mnm

zJRrdrrR

zJrR

zJ

rR

zJ

Rz kr

RzAJ(r)f

rfkrm

drd

rdrd

mnmn

mnmmn

;

0)(1

:function radialfor equation aldifferenti toReturning

22

2

2

2

PHY 711 Fall 2013 -- Lecture 26 1711/04/2013

r

RzAJ(r)f)(r 01

00101 ,

r

RzAJ(r)f)(r 02

00202 ,

520.5 406.20201 R

kR

k

PHY 711 Fall 2013 -- Lecture 26 1811/04/2013

cos

cos,

111

1111

rRzAJ

(r)f)(r

cos

cos,

121

1212

rRzAJ

(r)f)(r

016.7 832.31211 R

kR

k

PHY 711 Fall 2013 -- Lecture 26 1911/04/2013

Maple server evaluation project Request from Rick Matthews, Associate Provost

for Technology and Information Systems The University is exploring the possibility of

making Maple available on a server rather than on individual computersNeed students to exercise the software Need students to assess the performancNeed students to identify issues

Entirely voluntary

Possible pizza party or ??? rewards

PHY 711 Fall 2013 -- Lecture 26 2011/04/2013

Introduction to algebraic manipulation software

Available from the webpage: http://help.wfu.edu/public/vcl

If you have any trouble with this installation and setup, please contact Ching-Wan Yip at [email protected]

Note: This is a remote server that the university is testing. If you are off campus you will need to use VPN to gain access.

PHY 711 Fall 2013 -- Lecture 26 2111/04/2013

PHY 711 Fall 2013 -- Lecture 26 2211/04/2013

PHY 711 Fall 2013 -- Lecture 26 2311/04/2013