11/7/2018

1

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 1

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 30: Chap. 9 of F&W

Wave equation for sound in the linear approximation

1. Sound generation

2. Sound scattering

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 2

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 3

11/7/2018

2

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 4

01

2

2

22

tc

Solutions to wave equation:

22 where),(

:solution wavePlane

ckAet tii rkr

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 5

0 0

:onperturbatiin order lowest toEquations

0

0

vv

vfvv

v

tt

p

t

p

t applied

0 0

20 0

In terms of the velocity potential:

0

0 0

p p

t t

t t

v

v

v

Some comments about Monday’s lecture

0

0

Note that:

=

( )

0

, ,

for , , ' ( ')

t

pK t

t

t t

t t dt K t

p

t

r r

r r

v

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 6

2

00

0

),(

entropy (constant) thedenotes re whe),(

:density theof in terms pressure Expressing

cp

p

spp

sppspp

s

2

0 0

2 22

20 0

22 2 2

0 2

0 0

( ) 0

0 0

pc

t t

cc K t

t t t

ct t

Some comments about Monday’s lecture -- continued

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 7

Some comments about Monday’s lecture -- continued

vs

pc

ct

2

222

2

Here,

0

:airfor equation Wave

0

Boundary values:

ˆImpenetrable surface with normal moving at velocity :

ˆ ˆ ˆ

Free surface:

0 0 Φ

pt

n V

n V n v n

2

22 2

2

22 2

2

0

0

Additional relations:

0

ct

c

t

pt

pc p

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 8

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 9

'4

''

)','(

: thatshowcan We

rr

rr

rr

c

tt

ttG

Wave equation with source -- continued:

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 10

Derivation of Green’s function for wave equation

dett

ttttGtc

tti '

2

2

22

2

1)'(

thatRecall

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

rrrr

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 11

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 12

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

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 13

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 14

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

4

1

''4

1 : thatNote 2

BA

rrrr

R

e RG

ikR

4,

~

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 15

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

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 16

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 17

In order to solve an inhomogenous wave equation with a 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 solving equations with boundary conditions at infinity. For solving problems with surface boundary conditions where we know the boundary values or their gradients, the Green’s function must be modified.

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 18

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

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 19

3

2

S

If the integration volume includes the point ':

( , ) ' , ', '

ˆ ( ', ) ' , ' , ( ', ) '

V

V

G f d r

G G d r

r r

r r r r

r r r r r r n

3

2

S

3

2

S

( , ) ' ' , ,

ˆ ( , ) ' , ' , ( , )

Exchanging ':

( ', ) ' ' , ', '

ˆ ( ', ) ' , ' , ( ', ) '

V

V

G f d r

G G d r

G f d r

G G d r

r r r r r r

r r r r r r n

r r

r r r r r r

r r r r r r n

extra contributions from boundary

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 20

),(1

2

2

22 tf

tcr

Wave equation with source:

Example:

ˆ( , ) time harmonic piston of radius , amplitude

can be represented as boundary value of ( , )

f t a

t

r z

r

z

yx

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 21

'ˆ),'(~

','~

,'~

'),'(~

','~

,'~

),(~

2

S

3

rdGG

rdfGV

nrrrrrr

rrrr

' '

Note: Need Green's function with vanishing gradient at 0 :

' , where ' '; 04 ' 4 '

ik ik

z

e eG z z z

r r r r

r rr r r r

Treatment of boundary values for time-harmonic force:

222

222

0for

for 0~

:exampleour for aluesBoundary v

ayxai

ayx

zz

11/7/2018

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 22

''),'(

~,'

~),(

~

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 23

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 24

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

11/7/2018

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 25

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

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 26

z

yx

2

164320

2

sin

)sin(

2

1ˆ

:angle solidper power averaged Time

ka

kaJakcr

dΩ

dPe rj

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 27

Figure from Fetter and Walecka pg. 337

Scattering of sound waves –for example, from a rigid cylinder

11/7/2018

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 28

Scattering of sound waves –for example, from a rigid cylinder

( ) (

V

) ( ) ( )

elocity potential --

iinc sc inc e k rr r r r

2 2 22 2 2

2 22

2 2

Helmholz equation in cylindrical coordinates:

1 10

Assume:

1where ( ) 0

)

(im

m

m

m

k r kr rr r z

r

r

e R

d d mk R

dr r dr r

r r

r

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 29

cos( ) ( )i minc m

ikr im

m

e i e Je kr

k rr

( ) ( ) where Hankel function

represents an outgoing wave : ( ) ( ) ( )

Boundary condition at : 0

' ( )' ( ) ' ( ) 0

' ( )

imm m

m m m

r a

m m mm m

s

m m

cm

m

C kr

H kr J kr iN kr

r ar

J kai J ka C H ka C i

H k

H

a

e

r

11/7/2018 PHY 711 Fall 2018 -- Lecture 29 30

' ( )( ) ( )

' ( )scm imm

mmm

J kai kr

H kH

ae

r

/ 42(

Asymptotic form:

) i krmm

kr

H kri ekr

/ 4

/4

' ( )1 2( )

' ( )

' ( )2

' ( )

sci krikr imm

kr m

i

m

m

mm

m

eJ ka

f er H ka kr

J kaf

k H ka

e

e

r

11/7/2018

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11/7/2018 PHY 711 Fall 2018 -- Lecture 29 31

2df

d

For ka << 1

2 23 411 2cos

8k

df

da

/ 4' ( )2

' ( )m

i mm

m

J kaf

k H kae