PHY 711 Fall 2013 -- Lecture 29 111/11/2013

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 29 --

Chap. 9: Wave equation for sound

1. Standing waves

2. Green’s function for wave equation; wave scattering

PHY 711 Fall 2013 -- Lecture 29 211/11/2013

PHY 711 Fall 2013 -- Lecture 29 311/11/2013

PHY 711 Fall 2013 -- Lecture 29 411/11/2013

PHY 711 Fall 2013 -- Lecture 29 511/11/2013

Comments about exam

Note that this figure is misleading; a>b for physical case

12/cos2

2/sinsin

1sin

sin

sinsin

22

2

202

2

21

an

naab

nE

U

vv

m

bd

dd

d

db

d

dbb

d

d

T

0

2

2

sin2

1

sin

PHY 711 Fall 2013 -- Lecture 29 611/11/2013

Comment about exam – continued

mgkT

pss

dt

dp

s

gkT

ms

p

dt

sd

dt

dp

sgkTqVp

ms

pH

s

s

s

2

0

2

2

2

2

2

2

2

for 0 : thatNote

ln22

120

2

102

10

2

21

2

10 : thatSuppose

sms

p

ss

gkT

mss

p

dt

sd

tssts

PHY 711 Fall 2013 -- Lecture 29 711/11/2013

Comment about exam – continued

PHY 711 Fall 2013 -- Lecture 29 811/11/2013

Linearization of the fluid dynamics relations:

0 :equation Continuity

:motion ofequation Euler -Newton

v

fvvv

t

p

t applied

0

0

:mequilibriuNear

0

0

applied

ppp

f

vv

PHY 711 Fall 2013 -- Lecture 29 911/11/2013

0

:onperturbatiin order lowest toEquations

0

0

v

v

t

p

t

v :potentialVelocity

2

,, 00

:density theof in terms Pressure

cp

pps

0222

2

ct

PHY 711 Fall 2013 -- Lecture 29 1011/11/2013

v0

02

222

2

Here,

0

:airfor equation Wave

pp

c

ct

s

0 0

:surface Free

ˆ ˆ ˆ

:y at velocit moving ˆ normal with surface leImpenetrab

:aluesBoundary v

0

t

Φp

nvnVn

Vn

PHY 711 Fall 2013 -- Lecture 29 1111/11/2013

Time harmonic standing waves in a pipe

L

a

0222

2

ct

0 :surface freeAt

0ˆ :surface fixedAt

:aluesBoundary v

t

Φ

n

PHY 711 Fall 2013 -- Lecture 29 1211/11/2013

0),,,(11

11

)()()()()()(),,,(

:scoordinate lcylindricaIn

22

2

2

2

2

2

2

2

2

22

tzrkzrr

rrr

zrrr

rr

ezZFrRezZFrRtzr ikctti

ck

tc

:Define 01

2

2

22

PHY 711 Fall 2013 -- Lecture 29 1311/11/2013

0),,,(11 2

2

2

2

2

2

tzrkzrr

rrr

tiezZFrRtzr )()()(),,,(

integer)2()( ;)( mNFFeF im

nsrestrictioother plus real ;)( ziezZ

0)(1 22

2

2

2

2

rRk

r

m

dr

d

rdr

d

PHY 711 Fall 2013 -- Lecture 29 1411/11/2013

a

x'

dx

)(x'dJrJrR

dr

dR

rRr

m

dr

d

rdr

d

kk

mnmn

mnmm

ar

,0for where)()(

0 :conditionsboundary surfaceCylinder

0)(1

define For

22

2

2

2

22222

0)(1 22

2

2

2

2

rRk

r

m

dr

d

rdr

d

PHY 711 Fall 2013 -- Lecture 29 1511/11/2013

)( :functions Bessel xJm

m=0

m=1

m=2

PHY 711 Fall 2013 -- Lecture 29 1611/11/2013

m=0

m=1

m=2

)(

:sderviativefunction Besseldx

xdJm

Zeros of derivatives: m=0: 0.00000, 3.83171, 7.01559 m=1: 1.84118, 5.33144, 8.53632 m=2: 3.05424, 6.70613, 9.96947

PHY 711 Fall 2013 -- Lecture 29 1711/11/2013

Boundary condition for z=0, z=L:

...3,2,1 ,

sin)( 0)()0(

:pipeopen -openFor

pL

p

L

zpzZLZZ

p

22

2

2222

2

'

:sfrequencieResonant

L

p

a

xk

kc

ω

mnmnp

pmn

PHY 711 Fall 2013 -- Lecture 29 1811/11/2013

Example

05.3,84.1,00.0'

....42.9,28.6,14.3

'1

'22222

2

mn

mnmnmnp

x

p

p

x

a

L

L

p

L

p

a

xk

PHY 711 Fall 2013 -- Lecture 29 1911/11/2013

...3,2,1,0 ,

2

12

2

12cos)( 0)(

)0(

:pipe closed-openFor

pL

p

L

zpzZLZ

dz

dZ

p

Alternate boundary condition for z=0, z=L:

22

2

2

12'

L

p

a

xk mn

mnp

PHY 711 Fall 2013 -- Lecture 29 2011/11/2013

01

2

2

22

tc

Other solutions to wave equation:

22 where),(

:solution wavePlane

ckAet tii rkr

PHY 711 Fall 2013 -- Lecture 29 2111/11/2013

)'()'()','(1

where

)','()','(''),(

:function sGreen' of in termsSolution

),(1

2

2

22

3

2

2

22

ttttGtc

tfttGdtrdt

tftc

rrrr

rrrr

r

Wave equation with source:

PHY 711 Fall 2013 -- Lecture 29 2211/11/2013

'4

''

)','(

: thatshowcan We

rr

rr

rr

c

tt

ttG

Wave equation with source -- continued:

PHY 711 Fall 2013 -- Lecture 29 2311/11/2013

Derivation of Green’s function for wave equation

dett

ttttGtc

tti '

2

2

22

2

1)'(

thatRecall

)'()'()','(1

rrrr

PHY 711 Fall 2013 -- Lecture 29 2411/11/2013

Derivation of Green’s function for wave equation -- continued

2

2222 where','

~

:satisfymust ,~

,~

2

1,

,,~

:Define

ckGk

G

deGtG

dtetGG

ti

ti

rrrr

r

rr

rr

PHY 711 Fall 2013 -- Lecture 29 2511/11/2013

Derivation of Green’s function for wave equation -- continued

0,~

,~1

,~

:0for and ' Define --Check

'4,'

~

:'in isotropy assumingSolution

','~

22

222

'

22

RGkRGRdR

d

RRGk

RR

eG

Gk

ik

rr

rrrr

rr

rrrr

rr

PHY 711 Fall 2013 -- Lecture 29 2611/11/2013

Derivation of Green’s function for wave equation -- continued

R

eB

R

e ARG

BeA eRGR

RGRkRGRdR

d

RGkRGRdR

d

RRGk

R

ikRikR

ikRikR

,~

,~

0,~

,~

0,~

,~1

,~

:0For

22

2

22

222

PHY 711 Fall 2013 -- Lecture 29 2711/11/2013

Derivation of Green’s function for wave equation – continued need to find A and B.

4

1

''4

1 : thatNote 2

BA

rrrr

R

e RG

ikR

4,

~

PHY 711 Fall 2013 -- Lecture 29 2811/11/2013

Derivation of Green’s function for wave equation – continued

dee

dee

deGttG

ttii

ttiik

tti

c'

'

''

'

'42

1

'42

1

,'~

2

1','

rr

rr

rrrr

rr

rr

PHY 711 Fall 2013 -- Lecture 29 2911/11/2013

Derivation of Green’s function for wave equation – continued

'4

''

','

2

1 that Noting

'42

1',' '

'

rr

rr

rr

rrrr

rr

ctt

ttG

u δ dω eπ

dee

ttG

ui

ttii c

PHY 711 Fall 2013 -- Lecture 29 3011/11/2013

For time harmonic forcing term we can use the corresponding Green’s function:

'4

,'~ '

rrrr

rr

ike G

In fact, this Green’s function is appropriate for boundary conditions at infinity. For surface boundary conditions where we know the boundary values or their gradients, the Green’s function must be modified.

PHY 711 Fall 2013 -- Lecture 29 3111/11/2013

SV

rdhgghrdhggh

)g()h(2322 ˆ: thatNote

and functions woConsider t

theoremsGreen'

n

rr

)'(),'(~

),(~~~

22

22

rrrr

r

Gk

fk

rdGG

rdfG

Ggh

V

2

S

3

ˆ),(~

,'~

,'~

),(~

,,'~

'),(~

~ ;

~

nrrrrrr

rrrrrr

PHY 711 Fall 2013 -- Lecture 29 3211/11/2013

'ˆ),'(~

,'~

,'~

),'(~

',','~

),(~

2

S

3

rdGG

rdfGV

nrrrrrr

rrrr

PHY 711 Fall 2013 -- Lecture 29 3311/11/2013

),(1

2

2

22 tf

tcr

Wave equation with source:

),( of alueboundary v as drepresente becan

ˆ amplitude , radius ofpiston harmonic time),(

:Example

t

atf

r

zr

z

yx

PHY 711 Fall 2013 -- Lecture 29 3411/11/2013

'ˆ),'(~

','~

,'~

'),'(~

','~

,'~

),(~

2

S

3

rdGG

rdfGV

nrrrrrr

rrrr

0 ;'' re whe'4'4

,'~

:0at gradient ingith vanishfunction w sGreen' Need :Note''

zzzee

G

zikik

rrrrrr

rrrr

Treatment of boundary values for time-harmonic force:

222

222

0for

for 0~

:exampleour for aluesBoundary v

ayxai

ayx

zz

PHY 711 Fall 2013 -- Lecture 29 3511/11/2013

''),'(

~,'

~),(

~

0'

dydxz

GS: z

rrrr

0 ;'2

,'~

0 ;'' re whe'4'4

,'~

0'

'

0'

''

ze

G

zzzee

G

z

ik

z

ikik

rrrr

rrrrrr

rr

rrrr

PHY 711 Fall 2013 -- Lecture 29 3611/11/2013

a

z

ik

S: z

eddrrai

dydxz

G

0

2

0 0'

'

0'

'2'''

''),'(

~,'

~),(

~

rr

rrrr

rr

'sinsin''ˆ'

ˆcosˆsinˆ

plane; yz in the is ˆ Assume

'ˆ' ;For

2

rrr

rar

rrrr

zyr

r

rrrr

'sin''

'cos'' :domainn Integratio

ry

rx

PHY 711 Fall 2013 -- Lecture 29 3711/11/2013

aikr

iu

aikr

ikr

krJdrrr

eai

uJed

eddrrr

eai

0

0

0

2

0

'sin

0

2

0

'sinsin'

)sin'(''),(~

)('2

1 : thatNote

'''2

),(~

r

r

sin

)sin(),(

~

)()(

13

0

10

ka

kaJ

r

eai

wwJuuduJ

ikr

w

r

PHY 711 Fall 2013 -- Lecture 29 3811/11/2013

2

164320

2

*02

1

*21

sin

)sin(

2

1ˆ

:angle solidper power averaged Time

:average timeTaking

:fluxEnergy

ka

kaJakcr

dΩ

dP

i

p

p

e

e

e

rj

vj

vj