PHY 711 Classical Mechanics and
Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 28: Chap. 9 of F&W
Wave equation for sound considering
both linear and non-linear effects
1. Sound scattering
2. Non-linear effects in traveling
sound waves
3. Shocking effects
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
1
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
2
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
3
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
4
Solutions to linear wave equation in terms of velocity
potential:
1  
  2
0
2
c t
2
2
Plane wave solution :
 (r, t )  Ae
11/4/2015
ik r it
where
PHY 711 Fall 2015 -- Lecture 28
 
k  
c
2
2
5
Scattering of sound waves –
for example, from a rigid cylinder
Figure from Fetter and Walecka pg. 337
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
6
Scattering of sound waves –
for example, from a rigid cylinder
Velocity potential - inc (r )  eik r
 (r )   inc (r )   sc (r )
Helmholz equation in cylindrical coordinates:
1  
1 2
2
2
   k    r   0   r r r r  r 2  2  z 2  k    r 


2
2
Assume:   r  


m 
where
11/4/2015
eim Rm (r )
(no z dependence)
 d 2 1 d m2
2
 2  k  Rm (r )  0
 2
r dr r
 dr

PHY 711 Fall 2015 -- Lecture 28
7
 inc (r )  e
 sc (r ) 


m 
Cm eim H m (kr )
ik  r
e
ikr cos 



m 
i meim J m (kr )
where Hankel function
represents an outgoing wave : H m ( kr )  J m ( kr )  iN m (kr )
Boundary condition at r  a :
 i J 'm (ka )  Cm H 'm (ka )  0
m
11/4/2015

r
0
r a
J 'm (ka )
Cm  i
H 'm (ka )
PHY 711 Fall 2015 -- Lecture 28
m
8

J 'm (ka ) im
 sc (r )    i
e H m (kr )
H 'm (ka)
m 
m
Asymptotic form:
m
i H m (kr )
 sc (r )
 f  
kr 

kr 
2 i  kr  / 4
e
 kr

1 ikr
m J 'm ( ka ) im
e  i
e H m (kr )
r
H 'm (ka )
m 
2  J 'm (ka ) i  m  / 4
f    
e

 k m   H 'm (ka)
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
9
2
d
 f  
d
For ka << 1
2
d
1 3 4
2
 f     k a 1  2cos  
d
8
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
10
Now consider non-linear effects in sound
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
11
Effects of nonlinearities in fluid equations
-- one dimensional case
Newton - Euler equation of motion :
v
p
 v   v  f applied 
t


Continuity equation :
   v   0
t
Assume spatial variation confined to x direction ;
assume that v  vxˆ and f applied  0.
v
v
1 p
v 
t
x
 x


v
v

0
t
x
xPHY 711 Fall 2015 -- Lecture 28
11/4/2015
12
v
v 1 p
v 
0
t
x  x


v
v

0
t
x
x
Expressing p in terms of  : p  p (  )
p p 

p
2

 c ( )
where
 c2 ( )
x  x
x

For adiabatic ideal gas:
p
2  
c ( ) 
 c0  

 0 
2
11/4/2015
 1
dp
d

p

  
p  p0  
 0 
 p0
where c 
0
2
0
PHY 711 Fall 2015 -- Lecture 28
13

v
v c (  ) 
v 
0
t
x
 x


v
v

0
t
x
x
2
Expressing variation of v in terms of v  ρ  :
v 
v  c 2 (  ) 
v

0
 t
 x
 x


v 
v

0
t
x
 x
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
14
Some more algebra :
v  
  c 2 (  ) 
From Euler equation :
v
0


  t
x 
 x


v 
From continuity equation :
v
 
t
x
 x
v 
v   c 2 (  ) 
  
 
Combined equation :
0
 
 x 
 x
2
2
 v 
c ( )
   
2

  



 v  c 
0
t
x
11/4/2015
v
c



PHY 711 Fall 2015 -- Lecture 28
15
  
Assuming adiabatic process : c  c  
 0 
2
v dv
c


 d

 1
c 
2
0

 ' 
 v  c0   
0   0 
 1 / 2
2
0
γp0
0
d '
'
 1 / 2


2c0    
 
v
 1

  1   0 


  
 c  c0  
 0 
11/4/2015
 1 / 2
PHY 711 Fall 2015 -- Lecture 28
16
Summary :
dv
c

d



 v  c 
0
t
x
 1
  
γp0
2
Assuming adiabatic process : c  c  
c0 
0
 0 
 1 / 2
 1 / 2


  
2c0    
 
c  c0  
v
 1

  1   0 
 0 


2
11/4/2015
2
0
PHY 711 Fall 2015 -- Lecture 28
17
Traveling wave solution:
Assume :
   0  f ( x  u (  )t )
Need to find self - consistent equations for
propagation velocity u (  ) using equations


From previous derivations :
 v  c 
0
t
x
Apparently :
u( )  v  c
For adiabatic ideal gas and  signs :
   1     1/ 2

2

 
u  v  c  c0 

   1  0 
 1 


11/4/2015
PHY 711 Fall 2015 -- Lecture 28
18
Traveling wave solution -- continued:


 v  c 
0
t
x
Assume :    0  f ( x  u (  )t )   0  f ( x  v  c t )
For adiabatic ideal gas and  signs :
   1     1/ 2

2

 
u  v  c  c0 

   1  0 
 1 


Solution in linear approxiation :
  1
2 
  c0
u  v  c  v0  c0  c0 

  1  1 
 711 Fall 2015 -- Lecture 28

   0  f x  c0tPHY
11/4/2015
19
Traveling wave solution -- full non-linear case:
Visualization for particular waveform :
   0  f ( x  u (  )t )
Assume : f ( w)  f 0 s ( w)
f0

 1
s ( x  ut )
0
0
w
For adiabatic ideal gas :

u  c0 



u  c0 


11/4/2015
1   
 
 1  0 
 1 / 2
2 

 1 


f0
1 
1 
s ( x  ut ) 
 1  0

 1 / 2
PHY 711 Fall 2015 -- Lecture 28
2 

 1 

20
Visualization continued:
  1 / 2
   1

f0
2 

u  c0 
s ( x  ut ) 

1 
  1
0
 1



Plot s ( x  ut ) for fixed t , as a function of x :
Let w  x  ut
x  w  ut  w  u ( w)t  x ( w, t )
  1 / 2
   1

f0
2 

u ( w)  c0 
s ( w) 

1 
  1
0
 1



Parametric equations: plot s ( w) vs x( w, t ) for range of w
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
21
Linear wave:
Non-linear wave:
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
22
Linear wave
Non-linear wave
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
23
Analysis of shock wave
Plots of d
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
Solution becomes
unphysical
24
Analysis of shock wave -- continued
Before shock
t1
d1, dv1, dp1
Shock front
After shock
t2
d2, dv2, dp2
x
u
11/4/2015
PHY 711 Fall 2015 -- Lecture 28
25
Analysis of shock wave – continued
While analysis in the shock region is complicated, we
can use conservation laws to analyze regions 1 and 2
Assume  ( x, t )    x  ut 
p ( x, t )  p  x  ut 
v( x, t )  v  x  ut 
Continuity equation:
   v  u 
    v 

0
t
x
x
Conservation of energy and momentum:
 p2   2  v2  u   p1  1  v1  u 
2
 ò2 
11/4/2015
  v2  u   2   v1  u  1
2
1
p2
1
p1
2
2
v

u


ò

v

u

2 
1 
1
2
2
2
1
PHY 711 Fall 2015 -- Lecture 28
26
Analysis of shock wave – continued
For adiabatic ideal gas, also considering energy and
momentum conservation:
  1 p2
1
 2   1 p1
 1


1   1  p2   1
  1 p1
Velocity relationships:
 v1  u 
2
c12
where

 v2  u   1    1    1 p1 
p2 


1



1
  
  


2
p
c
2

p2 
1 
2


p
p
c12  1
and c22  2
1
2
2
1

2
For a strong shock:
 v1  u 
c12
11/4/2015
2
  1 p2

2
p1
 v2  u 
c22
2

  1
2
PHY 711 Fall 2015 -- Lecture 28
27
Analysis of shock wave – continued
For adiabatic ideal gas, entropy considerations::
p
Internal energy density: ε 
 CV T
  1 
1
First law of thermo: d   Tds  pd  


 1 
 p 
1 
p
ds   d 
  pd     CV d ln   


T      1  
  
 
 p 
s  CV ln      constant 
 
 p    
  p2 
  1
2
1
s2  s1  CV ln    
0 < s2  s1  CV  ln     ln 

 p1   2  
p


1


  1


11/4/2015
PHY 711 Fall 2015 -- Lecture 28
28