Waves on a Flat Surface of Discontinuity.ppt

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd
Sound Propagation
An Impedance Based Approach
Waves on a Flat Surface
of Discontinuity
Yang-Hann Kim
Chapter 3

Outline
3.1 Introduction/Study Objectives
3.2 Normal Incidence on a Flat Surface of Discontinuity
3.3 The Mass Law (Reflection and Transmission due to a Limp Wall)
3.4 Transmission Loss at a Partition
3.5 Oblique Incidence (Snell’s Law)
3.6 Transmission and Reflection of an Infinite Plate
3.7 The Reflection and Transmission of a Finite Structure
3.8 Chapter Summary
3.9 Essentials of Sound Waves on a Flat Surface of Discontinuity
2

3.1 Introduction/Study objectives
• What if we have a distributed impedance mismatch in space? How
does this change the propagation characteristics of waves in space?
• To begin with, the flat surface of a discontinuity in space, that is, a
wall, which creates an impedance mismatch in space is taken. We will
study how this mismatch transmits and reflects waves.
• This chapter begins with the simplest wall, which is modeled as a limp
wall. A limp wall is defined as one which has only mass.
• A more general wall creating an impedance mismatch is then
introduced.
3

• As illustrated in Figure 3.1, suppose that we have a flat surface of
discontinuity that separates two different media.
• Let us also assume that a wave propagates in the direction
perpendicular to the flat surface. We usually call this type of incident
wave to the surface “normal incidence” or “perpendicular incidence”.
4
Figure 3.1 The reflected and transmitted wave for a normal incident wave. (The subscripts denote the incident,
reflected, and transmitted wave, respectively. expresses the sound pressure with regard to time and space and
denotes the complex pressure amplitude)
wave
incident
wave
reflected
wave
transmitted
0
0
0
0
medium
c
ρ
Z  1
1
1
1
edium
m
c
ρ
Z 
x
)
x/c
(t
j
i
i e 0


 
P
p
0
j (t x/c )
r r e
p P 
 

)
/
(
P
p 0
c
x
t
j
t
t e 

 
, ,
i r t
p P

• The first point that we realize is that the pressure must be continuous
on the surface ( ); otherwise the surface will move according to
Newton’s second law. In addition, the velocity of a fluid particle at the
surface must also be continuous.
• First, the pressure continuity at can be written as
5
0
x 
0
x 
.
i r t
+ =
P P P
,
i r t
 
U U U
• The velocity continuity is expressed as
(3.1)
(3.2)
where the subscripts i, r, and t represent the incident, reflected, and
transmitted wave, respectively. P and U are the complex amplitude
of pressure and velocity.
P U
i r t

• The incident wave ( ) , reflected wave ( ) , and transmitted wave
( ) can therefore be written as
6
   
0
,
j t k x
i i
x, t e

 

p P
   
0
,
j t k x
r r
x, t e

 

p P
   
1
,
j t k x
t t
x,t e

 

p P
(3.3)
(3.4)
(3.5)
i
p r
p t
p
where k0 and k1 are defined
0
0
,
k
c


1
1
,
k
c


0
k 1
k
where c0 and c1 are the speed of sound in medium 0 and 1,
respectively.
0
c 1
c
(3.6)
(3.7)

• For a plane wave, we can rewrite Equation 3.2 as
7
0 0 1
,
i t
r
 
P P
P
Z Z Z
in which we use the relation where is the characteristic
impedance of the medium.
/

Z P U Z
• The ratio of to , that is, the reflection coefficient , can be
obtained from Equations 3.1 and 3.8:
1 0
1 0
.
r
i

 

Z Z
P
R
P Z Z
• The transmission coefficient, which is the ratio of to , can be
obtained from Equations 3.1 and 3.8 as
1
1 0
2
.
t
i
 

τ
P Z
P Z Z
(3.8)
(3.9)
(3.10)
r
P i
P R
t
P i
P

• To see how much power is essentially transmitted, we must determine
the velocity reflection and transmission coefficient. These can be
obtained by using the impedance relation of a plane wave, ,
from Equations 3.9 and 3.10. These are
8
/

Z P U
1 0
1 0
,
r
i



Z Z
U
U Z Z
0
1 0
2
.
t
i


U Z
U Z Z
velocity reflection coefficient :
velocity transmission coefficient :
(3.11)
(3.12)
• The power reflection/transmission coefficients are defined as the ratio
between the reflected/transmitted power and the power of the
incident wave.
* 2
1 0
2
*
1 0
1
2 .
1
2
r r
i i



PU Z Z
Z Z
PU
*
*
1 0
2
*
1 0
1
4
2 .
1
2
t t
i i


PU
Z Z
Z Z
PU
power reflection coefficient :
power transmission coefficient :
(3.13)
(3.14)

9
Figure 3.2 Pressure reflection and transmission
coefficient
where and
1 0 1 0
( ) ( )
  
R Z Z Z Z 1 1 0
(2 ) ( )
 
 Z Z Z
Figure 3.3 Velocity reflection and transmission coefficient
where and
1 0 1 0
( ) ( )
r i   
U U Z Z Z Z 0 1 0
(2 ) ( )
t i  
U U Z Z Z

• If the characteristic impedances of two media only have a real part
(e.g., water or air), then the power reflection and transmission
coefficients can be written as
• This can be derived from Equations 3.13 and 3.14. The sum of
transmitted and reflected power at the flat surface, that is, the
incident power, has to be 1, which can be derived by adding
Equations 3.15 and 3.16.
10
 
 
2
1 0
2
1 0
,
R

 

Z Z
Z Z
 
1 0
2
1 0
4
.

 

Z Z
Z Z
(3.15)
(3.16)

11
Figure 3.4 Change of the power reflection and transmission coefficients with regard to variation of the characteristic
impedances of the media, where and
   
2 2
1 0 1 0
R
   
Z Z Z Z  
2
1 0 1 0
(4 )

  
Z Z Z Z

3.3 The Mass Law
(Reflection and Transmission due to a Limp Wall)
• A limp wall is a wall that only has mass. In other words, the mass
effect is dominant compared to the stiffness or internal damping.
• “Locally reacting” assumption
The fluid particles only oscillate the mass particles that they are in
contact with, and the mass particles vibrate the fluid particles that
they contact.
12
Figure 3.5 Reflection and transmission due to the presence of a limp wall

• Let us first look at the forces acting on the unit surface area of a limp
wall.
• We assume that the wall harmonically oscillates, and then the
following force balance equation on has to be hold.
13
3.3 The Mass Law
0
x 
  2
i r t m ,

  
P + P P Y (3.17)
Where is mass per unit area (kg/m2).
• The velocity of the fluid particle on the left-hand side of the wall
should correspond to the vibration velocity of the wall, that is,
0 0
i r
j .

  
P P
Y
Z Z
(3.18)
• The velocity of the fluid particle on the right-hand side of the wall also
equal to the vibration velocity of the wall, that is,
(3.19)
0
t
j .

 
P
Y
Z
m

• We are interested in how much the waves are reflected and
transmitted relative to the incident wave amplitude. These ratios can
be obtained from Equations 3.17–3.19, which are
• The power transmission coefficient, which expresses how much
power is transmitted relative to the incident power, can be written as
• Substituting Equation 3.22 into Equation 3.21 gives
14
3.3 The Mass Law
0
,
2
r
i
j m
j m



 
 
P
R
P Z
0
0
2
,
2
t
i j m

 
 
τ
P Z
P Z
(3.20)
(3.21)
where R and t are the reflection and transmission coefficients,
respectively.
R 
* 2
*
1
2 .
1
2
t t
t
i
i i

PU
P
P
PU
(3.22)
 
   
2
2 0
2 2
0
2
.
2
m



τ
Z
Z
(3.23)

• Transmission loss ( or ) is defined as
15
3.3 The Mass Law
TL TL
R
10 2
1
10log .
TL
R


2
10
0
10log 1 .
2
TL
m
R

 
 
 
   
 
 
 
Z
(3.24)
(3.25)
• If is much greater than 1, then Equation 3.25 becomes
0
2
m

Z
10
0
20log dB.
2
TL
m
R

 
  
 
Z
(3.26)
• Mass law
The transmission loss increases by 6 dB as we double the frequency;
that is, 1 octave increase of frequency or mass per unit area is
doubled.
• From Equations 3.23 and 3.24, the limp wall transmission loss is

• To establish certain design guidelines, let us look at the frequency
that makes the transmission loss zero. We refer to this frequency as
“blocked frequency” ( ). This can be obtained from Equation 3.26,
because the blocked frequency must satisfy
16
3.3 The Mass Law
b
f
0
2
1.
2
b
f m


Z
(3.27)
Therefore,
0
.
b
f
m


Z
(3.28)
• For example, if is 415 (the characteristic impedance of air
at 20°C), then Equation 3.28 predicts the blocked frequency, that is,
0
Z
132
.
b
f
m
 (3.29)
2
/( )
kg m s

17
3.3 The Mass Law
6dB
6dB
2m
m
3dB
b
f 1
f 1
2 f
6dB
6dB
2m
m
3dB
b
f 1
f 1
2 f
Figure 3.6 Mass law: the graph shows that increases by 6 dB when the frequency doubles (1 octave). The mass law
is applicable from the blocked frequency
TL
R

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 18
3.3 The Mass Law
(Adapted from L. Cremer and M. Heckl, Structure-Borne Sound: Structural Vibrations and Sound Radiation
at Audio Frequencies, 2nd ed., Springer-Verlag ⓒ 1988, pp. 242.)
Table 3.1 Sound insulation materials and their density

• The sound on the left and right-hand sides of the wall is composed
of two different components: incidence and reflection on the left, and
trans-mission on the right.
• The pressure on the left side of the wall, , is found to obey
19
3.3 The Mass Law
2 .
i r i t
 
P + P P P (3.30)
• Using Equation 3.19, we can rewrite Equation 3.30 as
0
2 .
i r i j
 
P + P P YZ (3.31)
• The magnitude of the transmitted wave can be written as
0.
t j
 
P YZ (3.32)
i r
P + P

20
3.3 The Mass Law
Figure 3.7 The principle of superposition allows us to regard the incidence, reflection, and transmission phenomena as
the sum of the blocked and radiation pressure (blocked pressure and radiation pressure ).
2
b  i
P P 0
rad j
 
P YZ
• In summary, this observation leads us to conclude that the incident,
reflected, and transmission phenomena can be seen as the
superposition of the blocked pressure and the radiation pressure
induced by the wall’s motion (Figure 3.7).

• We now extend our understanding to more general cases. Figure 3.8
illustrates a partition that represents a more general flat surface of
discontinuity.
21
s
d
r
m
x
0
0
0
Z
0
medium
c

 0
0
0
Z
0
medium
c


wave
Incident
wave
d
Transmitte
wave
Reflected
)
/
(
P
p 0
c
x
t
j
i
i e 

 
)
x/c
(t
j
t
t e 0


 
P
p
)
/
(
P
p 0
c
x
t
j
r
r e 

 
d
r
s
s
d
r
m
x
0
0
0
Z
0
medium
c

 0
0
0
Z
0
medium
c


wave
Incident
wave
d
Transmitte
wave
Reflected
)
/
(
P
p 0
c
x
t
j
i
i e 

 
)
x/c
(t
j
t
t e 0


 
P
p
)
/
(
P
p 0
c
x
t
j
r
r e 

 
d
r
s
Figure 3.8 The reflection and transmission due to the partition (rd is linear damping coefficient and s is linear spring
constant)
d
r s

• To determine how much transmission will occur, we have to apply the
same laws that we used for the limp wall case: the velocity continuity
and the force balance between the wall and forces acting on the wall
of unit area.
• The transmission coefficient can be readily obtained as
22
   
0
0
2
.
/ 2 d
j m s r
 

   

Z
Z
(3.33)
- The imaginary part of denominator in Equation 3.33
- The real part of denominator in Equation 3.33
The mass contribution ( ) and the spring contribution
( )
have a 180 phase difference.
m
 /
s 

The first term expresses the radiation at both sides of the
wall, and the second term is what is lost by the linear
damping ( ).
d
r


23
Figure 3.9 Transmission and reflection, utilizing the concept of superposition ( , )
2
b i

P P 0
rad j
 
P YZ

• If we express Equation 3.33 using dimensionless parameters that can
be obtained by dividing every term by Z0, then we have
• The numerator (2) of Equation 3.34 essentially indicates that the
motion occurs in both directions. This leads us to rewrite Equation
3.34 as
24
0
Z
0 0 0
2
.
2 d
r
m s
j



   
   
   
   

Z Z Z
(3.34)
0 0 0
1
.
1
2 2 2
d
r
m s
j



   
   
   
   

Z Z Z
(3.35)

• We may also modify Equation 3.33 to understand the underlying
physics differently:
• This expression can be rewritten as
25
 
0
0
2
.
/ 2
d
j m jr s
 

   

Z
Z
,
f
p f


Z
Z + Z
where is the partition impedance and
is the fluid loading impedance in both directions.
 
/
p d
j m jr s
 
   
Z 0
2
f 
Z Z
(3.36)
(3.37)

26
Figure 3.10 (a) Normal incidence absorption coefficients. (b) Real and (c) Imaginary surface normal impedances of
some typical sound absorbing materials (thinsulate, foam, and fiberglass) (Data provided by Taewook Yoo and J.
Stuart Bolton, Ray W. Herrick Laboratories, Purdue University)

• We can obtain the transmission loss in a similar manner to that
described in Section 3.3, that is
• This is analogous to waves propagating along a string which is
attached to a mass-spring-dash pot system, as illustrated in Figure
3.11.
27
 
   
10 2
2 2
2 2 2
10 0 0 0
1
10log
1
10log 1 / / 2 / .
4
TL
d
R
m r
  

 
   
 
 

Z Z
(3.38)
m
s d
r
ave
incident w
wave
reflected
wave
d
transmitte
m
s d
r
m
s d
r
ave
incident w
wave
reflected
wave
d
transmitte
Figure 3.11 The incident, reflection, and transmission waves on strings that are attached to a single degree of freedom
vibration system (m is mass, s is linear spring constant, and rd is viscous damping coefficient)
where , which is the resonant frequency of the partition
in vacuum.
0 /
s m
 

• The case of :
28
0
 

• The case of :
0
 
• The case of :
0
 

10
0
20log .
2
TL
m
R

 
  
 
Z
10
0
20log .
2
TL
s
R

 
  
 
Z
10
0
20log 1 .
2
d
TL
r
R
 
 
 
 
Z
(3.39)
(3.40)
(3.41)
 The transmission loss is entirely dominated by the damping coefficient.
This implies that we have to increase the damping to make the transmission
loss larger.
 The transmission loss mostly depends on the linear spring constant.
 The transmission loss follows the mass law.

29
dB/octave
6
0



dB)
(
TL
R
dB/octave
6
0

 
0
/

frequency
natural region
law
mass
dB/octave
6
0



dB)
(
TL
R
dB/octave
6
0

 
0
/

frequency
natural region
law
mass
Figure 3.12 Transmission loss at a partition where is the undamped resonant frequency of the partition
0 /
s m
 

• If incident waves impinge on a flat surface of discontinuity with an
arbitrary angle other than , the fluid particles on the wall will
oscillate in directions both parallel and perpendicular to the wall (see
Figure 3.13).
30
/ 2

Figure 3.13 Reflection and transmission for oblique incidence ( : incidence, reflection, and transmission angle;
: incidence, reflection, and transmission wavelength)
, ,
i r t
   , ,
i r t
  

• The relation between wave number and wavelength
transforms Equation 3.43 to
• Suppose that an incident wave having pressure reaches a flat
surface of discontinuity at , where mediums 0 and 1 intersect. If
the incident wave reached ( ) after a period , arrives the surface ( )
at , then the following geometrical relations must hold:
31
( , )
i r t
p
0
x 
T
sin , sin , sin ,
i i r r t t
y y y
     
     
0
x 
(3.42)
where and are the incident, reflected, and transmitted angles,
respectively. and are the corresponding wavelengths.
,
i r
  t

, ,
i r
  t

• Equation 3.42 can then be written as
.
sin sin sin
i t
r
i r t
 

  
  (3.43)
 
2 /
k  

sin sin sin .
i i r r t t
k k k
  
  (3.44)
B A

• If the medium is non-dispersive, the dispersion relation must
be . Equation 3.44 can then be rewritten as
32
/
k c


0 0 1
sin sin
sin
.
i t
r
c c c
 

  (3.45)
• We can then deduce from Equation 3.45 that (the incident and
reflection angles are equal). Hence, Equation 3.45 can be rewritten as
i r
 

0 1
sin sin
.
i t
c c
 
 (3.46)
• To emphasize the characteristics of the media, let us denote and
as t and . Equation 3.46 can then written as
i
 t

0
 1

0 1
0 1
sin sin
.
c c
 
 (3.47)

• Similarly, we can write the relations
as , , . This is what we refer to as
Snell’s law.
• This law simply expresses that the wave number in the direction at
x=0 must be continuous in both media.
• Figures 3.14 and 3.15 depict the implications of these wave number
relations.
33
0 0 1 1
sin sin
k k
 
 0
i r
k k k
  1
t
k k

0
x 
Figure 3.14 Snell’s law expressed in the wave number
domain ( ). denotes the critical angle
0 1 0 1
;
k k c c
  crit

Figure 3.15 Snell’s law in the wave number domain
( )
0 1 0 1
;
k k c c
 
y

• These relations can also be obtained by considering what we already
observed in Section 3.2, that is, the pressure and velocity continuity
condition on the flat surface of discontinuity that has impedance
mismatch.
• We denote the incident, reflected, and transmitted waves as
34
   
   
   
, ,
, ,
, ,
i
r
t
j t k r
i i
j t k r
r r
j t k r
t t
r t e
r t e
r t e



  
  
  



p P
p P
p P
where
cos sin ,
cos sin ,
cos sin ,
i i i x i i y
r r r x r r y
t t t x t t y
k k e k e
k k e k e
k k e k e
 
 
 
 
  
 
where are the unit vectors in the direction, respectively.
,
x y
e e ,
x y
(3.48)
(3.49)

35
• From Equations 3.48 and 3.49, the pressure continuity at can be
written as
0
x 
sin sin
sin
.
i i t t
r r
jk y jk y
jk y
i r t
e e e
 

 
P P P
,
i r t
 
P P P
• Equation 3.50 can be simply written as
which is exactly the same as what was derived for the case of normal
incidence.
(3.50)
(3.51)

• The velocity continuity has to be satisfied by the fluid particles on the
surface of discontinuity, that is,
36
   
   
, , , , 0,
i r x t x
r t r t e r t e x
    
u u u
   
, ,
, , , ,
, ,
0,0,1 0,0,1 , ,
, .
i r t
j t k r
i r t i r t
i r t
i r t
k
r t e
c k


  
 
 

 
 
P
u
sin sin
0 0 0 0
sin
1 1
cos cos
cos
.
i i r r
t t
jk y jk y
i i i r r r
i r
jk y
t t t
t
k k
e e
c k c k
k
e
c k
 

 
 




P P
P
where
• Equations 3.49, 3.52, and 3.53 yield
(3.52)
(3.53)
(3.54)

• Equation 3.55 can also be rewritten as
37
0 0 0 0 0 0 1 1 1
.
/ cos / cos / cos
i t
r
c c c
     
 
P P
P
0,1 0,1
0,1
0,1
,
cos
c



Z
0 0 1
.
i t
r
 
P P
P
Z Z Z
we can rewrite Equation 3.56 as
• Equation 3.54 can be rewritten as
0 0 1
0 0 0 0 1 1
cos cos cos ,
i t
r
c c c
  
  
 
P P
P
(3.55)
where we adapted Equation 3.47 to simply emphasize the medium:
0
cos cos cos ,
i r
  
  1
cos cos .
t
 

(3.56)
• If we define the oblique wave impedance as
(3.57)
(3.58)

• The degree to which the wave is reflected and transmitted is
determined by the same approach as for the normal incident wave.
• Our findings are based on the assumption that the continuity on the
surface is independent of . Equations 3.50 and 3.54 are based on
this assumption. A surface of discontinuity that follows this
assumption is called a “locally reacting surface”.
• Let’s consider a special case as depicted in Figure 3.14. If the angle
of incidence is larger than the critical angle, then the transmission
angle has to be somewhat larger than 90° which is not physically
allowable. This situation can be modeled by writing the reflected
angle as
38
.
2
t j

 
  (3.59)
y

• By substituting Equation 3.59 into Equation 3.49 and then substituting
the result into Equation 3.48 we can obtain the reflected wave that is
exponentially decaying in the direction. We call this an “evanescent
wave”.
• In contrast to this case, if we consider that the speed of propagation
of medium 0 is larger than that of medium 1 (Figure 3.15), then the
transmitted angle cannot be larger than any critical angle.
39
x

3.6 Transmission and Reflection of an Infinite plate
• Suppose that the surface of the discontinuity does not locally react to
the waves, that is, incident, reflected, and transmitted waves as
illustrated in Figure 3.16.
• For simplicity, let us assume that we have plane waves.
40
y
x
i
P
t
P
r
P
i

r
 t

i

r

t

)
(
A
η y
k
t
j b
e 

 
0
0c

0
0c

y
x
i
P
t
P
r
P
i

r
 t

i

r

t

)
(
A
η y
k
t
j b
e 

 
0
0c

0
0c

Figure 3.16 Incident, reflected, and transmitted waves on the plate. The media are assumed to be identical

• The incident, reflected, and transmitted waves can be written as for
Equation 3.48, that is,
41
   
, ,
, , , ,
, .
i r t
j t k r
i r t i r t
r t e

  

p P
• The displacement of the surface of discontinuity ( ), a plate, only
propagates in the direction. We can therefore write the
displacement as

   
, ,
b
j t k y
y t e

 
 A

where is the amplitude of the displacement.
A
• The response of the plate to the pressures can then be written as

2 4
2 4 0
,
i r t x
m B p p p
t y 
 
   
 
 
where we assumed that the plate is thin enough to neglect the shear
effect.
(3.62)
(3.61)
(3.60)
y

• Equation 3.64 gives us the relation between the bending wave number
( ) and radiation frequency ( ), that is
• is the bending rigidity of the plate and can be written in this form:
42
B
 
3
2
,
12 1 p
Yd
B



2 4
0.
b
m Bk

  
where is the Poisson ratio, is the thickness of the plate, and is
Young’s modulus.
p
 d Y
• The characteristic equation that describes how the bending waves
generally behave can be obtained by substituting Equation 3.61 into
the homogeneous form of Equation 3.62. This yields
(3.64)
(3.63)
4 2
.
b
m
k
B

 (3.65)
b
k


• If we use the dispersion relation, then the speed of propagation of the
bending wave ( ) can be obtained as
43
1
2 4
.
b
b
B
c
k m
 
 
   
 
b
c
(3.66)
• Equation 3.66 simply states that the speed of propagation depends
on frequency. Note that a wave of higher frequency propagates faster
than a wave of lower frequency (see Figure 3.17). This characteristic
causes the shape of the wave to change as it moves in space.

44
1
ω 2
ω
1
b
k
2
1
4
1
ω
B
m
kb 






c
k


ω
k
2
2
2
1
1
1
b
b k
ω
c
k
ω
c 


0
t
t 
x
t
t
t 

 0
t
c 
2
t
c 
1
x
2
b
k
1
ω 2
ω
1
b
k
2
1
4
1
ω
B
m
kb 






c
k


ω
k
2
2
2
1
1
1
b
b k
ω
c
k
ω
c 


0
t
t 
x
t
t
t 

 0
t
c 
2
t
c 
1
x
2
b
k
Figure 3.17 Dispersive wave propagation in an infinitely thin plate

• If we express the velocity continuity mathematically, considering that
the velocity of the surface at the discontinuity ( ) has to be
exactly the same as the velocity of the fluid particle on the surface,
we arrive at
45
0
x 
sin sin
0 0 0 0
sin
0 0
/ cos / cos
.
/ cos
i i r r
b t t
jk y jk y
i r
i r
jk y jk y
t
t
e e
c c
j e e
c
 

   

 

  
A
P P
P
0
0
0 0
( ),
,
sin .
i r t
i r t
t
k k k
c
k k
   


  
  

(3.67)
• Equation 3.67 has to be valid for all y and, therefore, the exponents
have to be identical. This leads us to write
y
(3.68)

• Equation 3.67 can therefore be rewritten as
46
0 0
i r
j ,

  
P P
A
Z Z
0
t
j ,

 
P
A
Z
(3.69)
(3.70)
where , which is the oblique impedance of the
incident wave.
0 0 0 0
/ cos
c
 

Z
• Equations 3.60, 3.61, 3.62, 3.69, and 3.70 give us the transmission
loss of an infinite plate, that is
4
0 0
2
.
2
b
k B
m
j



 
  
 
 

Z Z
(3.71)
p
Z
4
.
b
p
k B
j m


 
  
 
 
Z (3.72)
• Equation 3.71 is a special case of Equation 3.37 when the partition
impedance ( ) is

• Recalling that and using Equation 3.68 once more,
we can rewrite Equation 3.73 as
• The magnitude of the transmitted wave ( ) can be obtained from
Equations 3.68 and 3.70, that is,
47
t
P
0 0
.
cos
t
t
j c




P A (3.73)
2
cos 1 sin
t t
 
 
0 0 2
0
.
1
t
b
j c
k
k

 
 
 
 
A
P
(3.74)
• Equation 3.60 can therefore be written as
     2
0 0
1 /
0 0 2
0
, .
1
b
b jk k k x
j t k y
t
b
r t j c e e
k
k



 
 
 
 
 
A
p (3.75)

• By setting and , Equation 3.75 can be rewritten as
48
0 0
/
k c

 /
b b
k c


     2
0 0
1 /
0 0 2
0
, .
1
b
b jk c c x
j t k y
t
b
r t j c e e
c
c



 
 
 
 
 
A
p
(3.76)
• The key feature of Equation 3.75 and 3.76 is expressed in the square
root of the propagation constant or the wave number in the direction.
If the propagation speed of the bending wave is smaller than the
speed of sound (subsonic), then we have an exponentially decaying
wave (an evanescent wave) in the direction. This means that there
are only waves in the vicinity of the plate and there is no transmission
(Figure 3.18).
x
x

49
Figure 3.18 The propagation characteristics of bending waves in subsonic, critical speed, and supersonic range

• We have studied the reflection and transmission of the discontinuity
of an infinite flat surface.
– They are relatively easy to tackle mathematically and therefore they are
easy to understand physically.
– The basic characteristics are often preserved for more practical cases, or
at least they provide a guideline to understand what would happen in
practical (more realistic) cases.
• For example, the mass law can be applied to any finite partition or
wall. Because it is a law for unit area, it can therefore be applied to a
finite structure as illustrated in Figure 3.19.
50

51
wall
limp
i
p
r
p
t
p

i
p
2

blocked pressure radiation pressure
partition
blocked pressure
i
p
r
p
t
p

i
p
2

radiation pressure
plate
finite
i
p
r
p
t
p

i
p
2

blocked pressure radiated pressure
wall
limp
i
p
r
p
t
p

i
p
2

i
p
r
p
t
p

i
p
2

partition
blocked pressure
i
p
r
p
t
p

i
p
2

radiation pressure
blocked pressure
i
p
r
p
t
p

i
p
2

radiation pressure
i
p
r
p
t
p

i
p
2

radiation pressure
plate
finite
i
p
r
p
t
p

i
p
2

i
p
r
p
t
p

i
p
2

Figure 3.19 Blocked pressure and radiation of a finite flat surface of discontinuity

• The edge effect makes the transmission with regard to unit area of the
partition larger than that of an infinite flat surface of discontinuity. We
can therefore express it as
52
,
f 

 
where and express the transmission coefficient of finite and
infinite flat surfaces of discontinuity, respectively.
f
 

• The transmission loss will therefore obey the relation:
, , .
TL f TL
R R 

(3.77)
(3.78)
,
TL
R 
• Equation 3.78 is very useful for practical applications. Because
over-estimates the transmission loss, it can be safely used as a
design value.

3.8 Chapter Summary
• We have examined how acoustic waves propagate in space and time
with regard to the characteristic impedance of the medium and the
driving point impedance.
• The limp wall example highlights that the mass law is the simplest
case and is a fundamental principle that can be used in practice. We
also found that the reflection and transmission can be regarded as
radiation due to the motion of the structure or partition.
• If we have oblique incidence, the reflection and transmission are
characterized by the surface of discontinuity.
• Transmission through a finite partition is due to the radiation of the
partition.
• Every transmission can be expressed in terms of partition and fluid
loading impedance.
53

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