Download 1.2 From String Vibration to Wave

Sound Propagation
An Impedance Based Approach
Chapter 1
Vibration and Waves
Yang-Hann Kim
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
Outline
• 1.1 Introduction/Study Objectives
• 1.2 From String Vibration to Wave
• 1.3 One-dimensional Wave Equation
• 1.4 Specific Impedance(Reflection and Transmission)
• 1.5 The Governing Equation of a String
• 1.6 Forced Response of a String: Driving Point Impedance
• 1.7 Wave Energy Propagation along a String
• 1.8 Chapter Summary
• 1.9 Essentials of Vibration and Waves
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.1 Introduction/Study Objectives
• Vibration can be considered as a special form of a wave (wave
propagations, Figure 1.1).
Figure 1.1 The first, second, and third modes of a string (demonstration by C.-S. Park and S.-H. Lee, 2005, at KAIST)
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• To understand how a wave propagates in space, let us start with the
simplest case.
Figure 1.2 Vibration of a string fixed at both ends (this demonstrates that the vibration can be expressed as the sum of two
modes: the second and third modes of the string)
• Figure 1.2 shows how two sinusoidal vibrations, whose frequencies are
f2 and f3, are actually composed of two different vibrations, that is, modes.
This can be mathematically expressed as
y ( x, t )  A2 sin
2 x
3 x
sin 2 f 2t  A3 sin
sin  2 f 3t    ,
L
L
(1.1)
where Ф represents the phase difference between the second and third
modes that are participating in the vibration.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• There is also a phase difference in space, as demonstrated by Figure 1.3.
Figure 1.3 How the second and third modes create the vibration
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• The first term of Equation 1.1 can be rewritten as
A2 sin
1   2 x

 2 x

2 x
 2 f 2t   cos 
 2 f 2t  .
 sin 2 f 2t  A2 cos 
2   L

 L

L
(1.2)
• Rearranging this equation in terms of xx gives
1 
2

2 x
A2 sin
sin(2 f 2t )  A2 cos
2 
L
L




L
2 
L

x

t

cos
x

t




(1/
f
)
L
(1/
f
)
2
2





(1.3)
L / (1// f2)
f 2 ) indicates a velocity that travels along the string.
where L/(1
• Equation 1.3 essentially means that there are two waves propagating
along the string in opposite directions with a velocity of L / T2 (T2  1/ f 2 ) .
• Similarly, the first or even n th mode can be interpreted in the same manner
as for the second and third mode cases.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• The string vibration can generally be written as

y ( x, t )   An sin
n 1
n
x sin  2 f nt   n  ,
L
(1.4)
where Φn is the phase of the nth mode.
• If we rewrite Equation 1.4 with respect to x, then

1 
n
y ( x, t )   An cos
L
n 1 2





n 
n 
2L
n 
2L
x

t


cos
x

t




 .
nTn
(n / L) 
L 
nTn
(n / L)  


(1.5)
• This equation essentially states the following: “There are cosine waves
propagating in the positive (+) and negative (−) directions with respect to
space, x.”
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• The general wave form, which is not simply a cosine wave, can be
mathematically expressed as
y( x, t )  g  x  ct   h  x  ct  ,
(1.6)
where g(•) and h(•) generally denote a wave form.
• Note that a wave g or h essentially depicts a wave form in arbitrary space
and time.
• These also propagate in space and time with the relation x+ct or x−ct.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• Figure 1.4 demonstrates how the function g moves along the axis x with
time. With respect to the x coordinate, we can now see how it changes in
time with respect to space.
Figure 1.4 The wave propagates in the positive (+)
(  ) x direction; g expresses the shape of the wave, c the wave propagation
speed, and tt and xx are the time and coordinate
• If we rewrite the function or wave g with regard to time, then we obtain
  x 
g  x  ct   g  c  t    .
  c 
(1.7)
• Equation 1.7 states that the right-going wave in space can be seen as the
wave propagates in time.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• Figure 1.5 essentially illustrates that what we can see in space is related to
what we observe in time; this graph is typically referred to as a wave
diagram.
Figure 1.5 Wave diagram: waves can be observed at the x coordinate (space) and t axis (time), where Δt denotes infinitesimal
time, x1,2 and t1,2 indicates arbitrary position and time, and y is the wave amplitude
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• The sine wave is a special wave that can be expressed by Equation 1.7.
The sine wave, propagating to the right, is expressed
y  x, t   Y sin  k  x  ct    
(1.8)
where k converts the units of the independent variable of the sine function
to radians; x and ct are in units of length; Y represents amplitude and Φ is
an arbitrary phase.
• We rewrite Equation 1.8 as
y( x, t )  Y sin   kx  kct      Y sin  kx  t    ,
(1.9)
where kc   .
• It relates the variable that expresses the changes of space (x), k, with that
related to time (t), ω. That is,
k
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
c
Yang-Hann Kim
(1.10)
.
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• Equation 1.10 can be rewritten in terms of frequency (cycles/sec, Hz), or
period (sec), that is
k

c

2 f
1
 2
.
c
cT
(1.11)
• We can rewrite Equation 1.11 as
k
2

(1.12)
,
where k represents the number of waves per unit length (1/ ). We call this
the wave number or a propagation constant.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• Note that the distance across which a wave travels for a period T with a
propagation speed c will be a wavelength (λ) (see Figure 1.6).
Figure 1.6 Waves can be seen for one period: T is period (sec), c is propagation speed (m/sec), and x and t represent the space
and time axis, respectively
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.2 From String Vibration to Wave
• We can also obtain an additional relation from Equations 1.10 and 1.12.
That is,
c
 .
(1.13)
f
• This states that the variables which express space (λ) and time (ff ) are not
independent of each other.  “dispersion relation”
• By using a complex function, Equation 1.9 can be rewritten as

y( x, t )  Im Ye
j  kx t  
  ImYe  ,
j kx t
(1.14)
where Y is the complex amplitude.
• For the sake of simplicity, Equation 1.14 will be written as
y( x, t )  Ye
j  kx t 
.
(1.15)
• We can also express Equation 1.15 with respect to time instead of space,
that is
 j t  kx 
y ( x, t )  Ye 
.
(1.16)
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.3 One-dimensional Wave Equation
• Any one-dimensional wave can be expressed as
y ( x, t )  g  x  ct   h  x  ct  .
(1.17)
• We would like to determine the derivative of Equation 1.17 with regard to
time and space and thereby examine its underlying physical meaning.
• Let’s see how Equation 1.17 behaves in the case of a small spatial change:
y
(1.18)
 g ' h ',
x
where ' denotes the derivative of each function with respect to its
arguments (e.g., g ( z )'  dg / dz ). Its time rate of change is expressed as
y
 cg ' ch '
t
which leads to
Sound Propagation: An Impedance Based Approach
g
g
 cg '  c ,
t
x
h
h
 ch '  c .
t
x
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
(1.19)
(1.20)
(1.21)
15
1.3 One-dimensional Wave Equation
• Figure 1.7 illustrates the associated kinematics of the right-going and leftgoing wave.
Figure 1.7 Understanding waves from the perspective of wave kinematics (a wave that has a positive slope or negative
slope has a negative or positive rate of change, i.e., velocity)
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.3 One-dimensional Wave Equation
• If we differentiate Equations 1.20 and 1.21, we obtain
2
2 g
2  g
c
,
2
2
t
x
h
2  h

c
.
2
2
t
x
2
2
(1.22)
• Any one-dimensional wave ( y ( x, t ) ) which has left-going and right-going
waves with respect to the selected coordinates satisfies the partial
differential equation:
2
2 y
2  y
c
.
2
2
t
x
(1.23)
• Equation 1.23 can then be rewritten as
2 y 1 2 y
 2 2.
2
x
c t
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
(1.24)
17
1.3 One-dimensional Wave Equation
• A three-dimensional version of Equation 1.24 can be written as
1  2
 2 2,
c t
2
(1.25)
where  ( x, y, z, t ) denotes the amplitude of three-dimensional wave.
• The boundary condition can generally be written as
  

 ,
x
(1.26)
where ψ expresses the general force acting on the boundary. α and β are
coefficients that are proportional to force and spatial change of force,
respectively.
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.3 One-dimensional Wave Equation
• Two types of boundary conditions: passive and active
Figure 1.8 Examples of passive and active boundary conditions
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.4 Specific Impedance (Reflection & Transmission)
• Waves traveling along a string are representative of the many possible
one-dimensional waves.
• Let us first examine waves propagating along two different strings, as
illustrated in Figure 1.9.
Figure 1.9 Waves in two strings of different thickness (g1 is an incident wave, h1 represents a reflected wave, and g2 is a
transmitted wave)
• We wish to determine the relation between the incident wave g1, the
reflected wave h1 and the transmitted wave g2.
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.4 Specific Impedance (Reflection & Transmission)
• Let’s envisage what really happens at this discontinuity, and then express
it mathematically.
• The velocities in the y direction (uy) of the thin string and thick string have
to be identical. In addition, the resultant force in the y direction (fy) has to
be balanced according to Newton’s second law. These two requirements
at the discontinuity are expressed mathematically as
u y 
f y 
x  0
x  0
 u y 
x  0
,
 f y    0.
x 0
(1.27)
(1.28)
• Denote the waves on the negative x axis region, #1 string, as y1 and
express the wave that propagates in the positive x axis as y2. Describing
these waves with regard to time, they can be written as


x
x
y1  g1  t    h1  t   ,
 c1 
 c1 
(1.29)

x
y2  g 2  t   .
 c2 
(1.30)
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.4 Specific Impedance (Reflection & Transmission)
• The velocity in the y direction at x  0 can be written as
u y 
x  0
• At x  0 , it is

u y 
y1 
'
'



g

h
1  x  0
1  x  0 .

t  x 0
(1.31)
y2 
'
  .

g
2
x 0
t  x 0
(1.32)
x  0

• We therefore obtain the following equality since the velocity must be
continuous:
g1' 
 h1' 
 g2'  .
(1.33)
x 0
x 0
x 0



• The forces in the y direction ( f y ) are related to the tension along the string
aTL and the slope (Figure 1.10) as
f y 
x  0
 TL
y
,
x
f y 
x  0
 TL
y
.
x
(1.34)
• Therefore, we can rewrite Equation 1.28 as
TL '
T
T
g1  x 0  L h1'  x 0  L g 2'  x 0 .
c1
c1
c2
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© 2010 John Wiley & Sons (Asia) Pte Ltd
(1.35)
22
1.4 Specific Impedance (Reflection & Transmission)
Figure 1.10 Forces acting on the end of the string where TL is tension, fy describes the force in the y direction, y indicates the
amplitude of the string, and x denotes the coordinate)
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.4 Specific Impedance (Reflection & Transmission)
• We can postulate that the string’s wave amplitude at x  0, t  0 is zero. We
can therefore write Equations 1.33 and 1.35 as
(1.36)
g '1  0   h '1  0   g '2  0  ,
TL
T
T
g '1  0   L h '1  0   L g '2  0  .
c1
c1
c2
(1.37)
• The ratio of the string’s force in the yy direction (fy) and the associated
velocity (uy) can be written as
fy
uy

TL
.
c
(1.38)
• The force that can generate the unit velocity is generally defined as the
impedance.
• We normally express this using the complex function Z, which allows us to
express any possible phase difference between the force and velocity.
Therefore, Equation 1.37 can be rewritten as
Z1 g '1  0   Z1h '1  0   Z 2 g '2  0 
where Z1 and Z2 are equal to TL / c1 and TL / c2, respectively.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
(1.39)
24
1.4 Specific Impedance (Reflection & Transmission)
• Using Equations 1.36 and 1.39, the reflection ratio (h1/g1) can be
expressed as
h1  0  Z1  Z 2

.
g1  0  Z1  Z 2
(1.40)
• The transmission ratio (h1/g1) can be written as
g2  0
2 Z1

.
g1  0  Z1  Z 2
(1.41)
• The ratio of the reflected wave and transmitted wave to the incident wave
depends entirely on the string’s impedance, TL / c .
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.4 Specific Impedance (Reflection & Transmission)
• Figure 1.11 exhibits how the waves on a string propagate when they meet
a change of impedance or, in this case, a change of thickness of string.
Figure 1.11 Incident, reflected, and transmitted waves on a string; note the phase changes of the reflected and transmitted
waves compared to the incident wave. The thin line has impedance Z1 and the thick line has impedance Z 2
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.5 The Governing Equation of a String
• Let us examine an infinitesimal element of string (Figure 1.12).
Figure 1.12 Newton’s second law on an infinitesimal element of a string (notation as for Figure 1.10)
• Newton’s second law in the x direction can be written:
2 x
TL cos  TL  dTL  cos   d    L ds 2 .
t
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(1.42)
27
1.5 The Governing Equation of a String
• The force and motion in the y direction can be written:
2 y
TL sin   TL  dTL  sin   d    L ds 2 ,
t
(1.43)
where  expresses the slope of the string with respect to the x axis at an
arbitrary position of x:
tan  
y
.
x
(1.44)
• The change of this slope with regard to a small change in x (dx) can be
written as
y  2 y
  d   2 dx
x x
(1.45)
using a Taylor expansion.
• Assuming that the displacement of the string is small enough to be
linearized, then
sin    ,
(1.46)
cos  1.
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.5 The Governing Equation of a String
• Equations 1.42 and 1.43 thus become
2 x
TL  TL  dTL    L ds 2 ,
t
(1.47)
 y  2 y 
y
2 y
TL  TL  dTL    2 dx    L ds 2 .
x
t
 x x

(1.48)
• The small ds can be rewritten as
ds 
 dx    dy 
2
2
2
 1  y 2 
 y 
 dx 1     dx 1     .
 2  x  
 x 


(1.49)
• Its square can therefore be neglected compared to other variables.
Therefore, we can approximate
(1.50)
ds  dx.
• The small change of tension dTL can be expressed by a first-order
approximation as
T
(1.51)
dTL  L dx.
x
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.5 The Governing Equation of a String
• Equation 1.47 can be rewritten as
TL
2 x
 L 2 .
x
t
(1.52)
• We can easily write Equation 1.48 as
2 y
2 y
TL 2   L 2 .
x
t
(1.53)
• Rearranging Equation 1.53 results in
2 y L 2 y

.
x 2 TL t 2
(1.54)
• Equation 1.54 can be summarized as
2 y 1 2 y

,
x 2 cs2 t 2
cs2 
TL
L
.
(1.55)
(1.56)
where cs is the propagation speed of the string.
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.5 The Governing Equation of a String
• Recall that the impedance of the string Z is
Z=
TL
.
cs
(1.57)
• Using Equation 1.56, we can rewrite Equation 1.57 as
Z = Lcs .
(1.58)
• Impedance has two different implications.
- The impedance is a measure of how effectively the force can generate
velocity (response), that is, the input and output relation between force and
velocity.
- The impedance represents the characteristics of the medium.
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
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1.6 Forced Response of a String: Driving Point Impedance
• We first investigate what happens if we harmonically excite one end of a
semi-infinite string.
Figure 1.13 Wave propagation by harmonically exciting one end of a semi-infinite string (T is period, cs is propagation speed, λ is
the wavelength, f is the frequency in Hz (cycles/sec), and ω is the radian frequency in rad/sec)
Sound Propagation: An Impedance Based Approach
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© 2010 John Wiley & Sons (Asia) Pte Ltd
32
1.6 Forced Response of a String: Driving Point Impedance
• For mathematical convenience, we begin by expressing the waves in
Figure 1.13 using a complex function:
y  x, t   g  x  cst .
(1.59)
• The boundary condition at x  0 can be written as
y  0, t   g  cst   Ye  jt ,
(1.60)
 jt
where Ye denotes the response of the string due to the excitation ( Fe jt )
at x  0.
• We can therefore rewrite Equation 1.60 as
g  cst   Ye
jk   cs t 
,
(1.61)
where we use the dispersion relation k   / cs .
• If we rearrange Equation 1.61 using an independent variable  , then we
obtain
(1.62)
g    Ye jk .
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
33
1.6 Forced Response of a String: Driving Point Impedance
• We can therefore substitute α by x  cst , which gives us
g  x  cst   Ye
 jk  x cs t 
 Ye
 j t  kx 
.
(1.63)
• The velocity can be expressed using Equation 1.60:
uy  0, t  
y 
 jt


j

Y
e
.
t  x 0
(1.64)
• The force at the end of the string is related to the tension and the slope of
string (Figure 1.10):
f y  0, t  = Fe jt  TL
y 
.

x  x 0
(1.65)
• We can rewrite the impedance at the end as
Zm0 
f y  0, t 
uy  0, t 
  L cs .
(1.66)
• The characteristics of the driving point impedance determine the spatial
phenomenon of wave propagation, that is, the ways in which waves
propagate in space.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
34
1.6 Forced Response of a String: Driving Point Impedance
• Another extreme case that can demonstrate how the driving point
impedance reflects the wave propagation along a string is a string that has
finite length L.
• One end (x=0) is harmonically excited and the other end (x=L) is fixed.
x, t 
• The boundary condition at x=L requires that the displacement y y(x,t)
always be 0. The solution that satisfies the governing wave equation and
this boundary condition can be written as
y  x, t   Y sin k  L  x  e  jt .
(1.67)
0 , then we have
• If we calculate the velocity using Equation 1.67 at xx =0
uy  0, t  
y 
 jt


j

Y
sin
kL
e
.

t  x 0
(1.68)
 0 is
• The force at xx=0
f y  0, t  = TL
Sound Propagation: An Impedance Based Approach
y 
 jt

T
kY
cos
kL
e
.
L

x  x 0
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
(1.69)
35
1.6 Forced Response of a String: Driving Point Impedance
• Equations 1.68 and 1.69 give us the impedance (specifically, the driving
point impedance Zm0) at x  0 . That is,
Z m0 
f y  0, t 
uy  0, t 
j
TL
cot kL  j  Lcs cot kL.
cs
(1.70)
• When the wavelength is large compared to the length of the string, then
Equation 1.70 reduces to
Z m 0  j  L cs
1
.
kL
(1.71)
• Rearranging this equation, we obtain
Zm0  j
TL
.
L
(1.72)
• Driving point impedance represents how much force is required to obtain
unit velocity, or how much velocity will be generated by a unit force at the
point of interest.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
36
1.6 Forced Response of a String: Driving Point Impedance
Figure 1.14 The driving point impedance of a finite string (k is wave number and L is the length of the string)
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
37
1.6 Forced Response of a String: Driving Point Impedance
• Summary of Driving point impedances
Table 1.1 Driving point impedances
Nomenclature: ρL : mass per unit length of string, rod; ρA : mass per unit area of membrane; ρ : mass per volume of plate;
λP : Poisson’s ratio; cs : speed of propagation of string; cb  Y /  ; c p  Y /  (1  p2 ) ; ω : angular frequency; k : wavenumber;
L : length of string, rod, and bar; Y : Young’s modulus; S : cross-sectional area of rod and beam; χ : radius of gyration of
beam and plate; d : thickness of plate; Tm : membrane tension (N/m); vb : propagation speed of bar (=  cb  , depending
 c p  , depending on frequency); Z mM0 : driving point impedance by
on frequency); vp : propagation speed of plate (aaaaaaa
F
bending moment of beam; Z m 0 : driving point impedance by shear force of beam and plate.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
38
1.7 Wave Energy Propagation along a String
• Let’s determine how much energy can be stored in an infinitesimal
element of string.
Figure 1.15 The change of an infinitesimal element of a string in infinitesimal time
• The kinetic and potential energy in the infinitesimal element of the string
can be written
1
 y 
dEK   L dx    ,
2
 t 
2
dEP  TL
 dx  dy  
2
2
(1.73)
1
 y 
 dx  TL dx   ,
2
 x 
2
(1.74)
where d expresses a small element.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
39
1.7 Wave Energy Propagation along a String
• The total energy of the string can be written
1
  y 
 y 
dE    L    TL  
2
 x 
  t 
2
2


 dx.


(1.75)
• Energy density can be expressed by

dE
.
dx
(1.76)
• The total energy in the string can be written as
E
1
 dx   Lcs2 
2
 y 2  1 y 2 
   
  dx.
 x   cs t  
(1.77)
• Equation 1.77 demonstrates that the greater the slope along the string
(with regard to x ) and the faster the speed of wave propagation, the more
energy we have.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
40
1.7 Wave Energy Propagation along a String
• Consider that we raise one end of the string (see Figure 1.16).
• The kinetic energy can be approximated as
• The potential energy is
2
1  uy 
TL   cE
2  cE 
1
2
 L  cE  u y2
2
.
; this can be readily obtained by the work
done due to the elongation of string.
Figure 1.16 Energy propagates along a string by raising one end ( TL is tension along the string, cE energy propagation speed, 
time, and u y lifting velocity at the end)
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
41
1.7 Wave Energy Propagation along a String
• These lead to the equation:
2
1
1 u 
TL u y   L  cE  u y2  TL  y  cE ,
cE
2
2  cE 
uy
(1.78)
which gives us
cE2 
TL
L
.
(1.79)
• The speed of energy propagation is identical to the phase velocity of a
string.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
42
1.8 Chapter Summary
• We have studied wave propagation along a piece of string, which is a
typical one-dimensional wave.
• A wave is an expression of a space–time relation.
• A harmonic wave solution gives us the dispersion relation, which
determines the relation between wave number and frequency and is
determined by the characteristics of the medium.
• The ways in which waves are reflected and transmitted are completely
determined by the characteristic impedances of two strings, which create
an impedance mismatch between the strings.
• The driving point impedance represents how the waves on a string
propagate.
Sound Propagation: An Impedance Based Approach
Yang-Hann Kim
© 2010 John Wiley & Sons (Asia) Pte Ltd
43