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


2

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)


3

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 f

2 and f

3

, are actually composed of two different vibrations, that is, modes.

This can be mathematically expressed as

( , )



A

2 sin

2



L x sin 2

 f t

2



A

3 sin

3

 x

L

  f t

3

  

,

(1.1) where Ф represents the phase difference between the second and third modes that are participating in the vibration.

4



• 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


5


• The first term of Equation 1.1 can be rewritten as

A

2 sin

2

 x

L

 sin 2

 f t

2



1

2

A

2

 cos

2

 x



2



L f t

2

 cos

2

 x



2



L f t

2



.

• Rearranging this equation in terms of x x gives

A

2 sin

2

 x sin(2

 f t

2

)

L



1

2

A

2

 cos

2



L



 x



L

(1 / f

2

) t





 cos

2



L



 x



L

(1 / f

2

) t







 where

L / ( 1 / f2) ) indicates a velocity that travels along the string.

(1.2)

(1.3)

• Equation 1.3 essentially means that there are two waves propagating along the string in opposite directions with a velocity of

L T T

2 2



1/ f

2

)

.

• n as for the second and third mode cases.

6



• The string vibration can generally be written as





 n



1

A n sin n



L x where Φ n is the phase of the n th mode.

  f t n

  n



, (1.4)

• If we rewrite Equation 1.4 with respect to x

, then

 n







1

1

2

A n

 cos

 n



L



 x



2 L nT n t



 n

( n



/ )





 cos n



L



 x



2 L nT n t



 n

( n



/ )







.

(1.5)

• This equation essentially states the following: “There are cosine waves propagating in the positive

( + ) and negative

(

−

) directions with respect to space, x

.”

7



• The general wave form, which is not simply a cosine wave, can be mathematically expressed as

( , )

      

, (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

.


8


• 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 speed, and t t and x are the time and coordinate

(+) x direction; g expresses the shape of the wave, c the wave propagation

• If we rewrite the function or wave g with regard to time, then we obtain

  ct

  g







 x c 





.

(1.7)

• Equation 1.7 states that the right-going wave in space can be seen as the wave propagates in time.

9



• 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 Δ t denotes infinitesimal time, x

1,2 and t

1,2 indicates arbitrary position and time, and y is the wave amplitude


10


• 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 sin

     



(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 sin

  kx kct

 

Y sin

 kx

  t

  

, where kc

  .

(1.9)

• It relates the variable that expresses the changes of space

( x )

, k

, with that related to time

( t )

, ω . That is, k





.

c

(1.10)

11



• Equation 1.10 can be rewritten in terms of frequency (cycles/sec, Hz), or period (sec), that is k







2

 c c f



2



1 cT

.

(1.11)

• We can rewrite Equation 1.11 as k



2





, (1.12) where k represents the number of waves per unit length ( the wave number or a propagation constant.

1/

 ). We call this


12


• 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


13


• We can also obtain an additional relation from Equations 1.10 and 1.12.

That is,

  c

.

f

(1.13)

• This states that the variables which express space ( independent of each other.

 “dispersion relation”

λ ) and time ( f f

) are not

• By using a complex function, Equation 1.9 can be rewritten as

( , )



Im



Ye

  t







Im



Y e

   t





,

(1.14) where

Y is the complex amplitude.

• For the sake of simplicity, Equation 1.14 will be written as

( , )



Y e

   t



.

(1.15)

• We can also express Equation 1.15 with respect to time instead of space, that is

( , )



Y e

 j

  t kx



.

(1.16)

14


1.3 One-dimensional Wave Equation

• Any one-dimensional wave can be expressed as

( , )

      

.

(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)

 x

  h ', where

' denotes the derivative of each function with respect to its arguments (e.g.,

( )'



/

). Its time rate of change is expressed as

 y

 t

  cg '

 ch ' which leads to

 g

 t

 h

 t

  cg '

  c

 ch '

  c

 g

 x

 h

 x

.

,


(1.19)

(1.20)

(1.21)

15


• 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)


16


• If we differentiate Equations 1.20 and 1.21, we obtain

 2 g

 t

2

 c

2

 2 g

 x

2

,



2 h

 t

2

 c

2



2 h

 x

2

.

(1.22)

• Any one-dimensional wave ( differential equation: y x t

) which has left-going and right-going waves with respect to the selected coordinates satisfies the partial



2 y

 t

2

 c

2



2

 x

2 y

.

(1.23)

• Equation 1.23 can then be rewritten as





2 x y

2

 c

1

2





2 t

2 y

.

(1.24)

17



• A three-dimensional version of Equation 1.24 can be written as where 



1

 2

 c

2

 t

2

, denotes the amplitude of three-dimensional wave.

(1.25)

• 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.


18


• Two types of boundary conditions: passive and active

Figure 1.8 Examples of passive and active boundary conditions


19

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 ( g

1 transmitted wave) is an incident wave, h

1 represents a reflected wave, and g

2 is a

• We wish to determine the relation between the incident wave g

1 reflected wave h

1 and the transmitted wave g

2

.

, the

20



• Let’s envisage what really happens at this discontinuity, and then express it mathematically.

• The velocities in the y direction ( u y

) of the thin string and thick string have to be identical. In addition, the resultant force in the y direction ( f y

) has to be balanced according to Newton’s second law. These two requirements at the discontinuity are expressed mathematically as u y x



0



 u y x



0



, (1.27) f y x



0



 f y x



0





0 .

(1.28)

• Denote the waves on the negative x axis region, #1 string, as y express the wave that propagates in the positive x axis as y these waves with regard to time, they can be written as

2

1 and

. Describing y

1

 g t

1







 x c

1





 h t

1

 



  x c

1







, y

2

 g

2





 t

 x c

2







.

(1.29)

(1.30)

21



• The velocity in the y direction at x



0

 can be written as u y

 x



0



 y

1

 x



0



 g

1

' x



0



 h

1

' x



0



.

(1.31)

• At x



0

 , it is u y

 x



0



 y

2



 g

2

' x



0



.

(1.32) x



0



• We therefore obtain the following equality since the velocity must be continuous: g

1

' x



0



 h

1

' x



0



 g

2

' x



0



.

(1.33)

• The forces in the

T

L y direction ( ) are related to the tension along the string a and the slope (Figure 1.10) as f y

 x



0



 

T

L

 y

 x

, f y

 x



0





T

L

 y

 x

.

(1.34)

• Therefore, we can rewrite Equation 1.28 as

T

L g

1

' c

1 x



0





T

L c

1 h

1

' x



0





T

L c

2 g

2

' x



0



.

(1.35)

22



Figure 1.10 Forces acting on the end of the string where

T

L amplitude of the string, and x denotes the coordinate) is tension, f y describes the force in the y direction, y indicates the


23


• We can postulate that the string’s wave amplitude at can therefore write Equations 1.33 and 1.35 as g

1

   h

1

   g ' 0 ,

2

  x



0, t



0 is zero. We

(1.36)

T

L g

  

T

L h

  c

1 c

1

• The ratio of the string’s force in the velocity ( u y

) can be written as



T

L g

  (1.37) c

2 y y direction ( f y

) and the associated f y 

T

L u y 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

Z g ' 0

1 1



Z h ' 0

1 1

  

Z g ' 0

2 2

  (1.39) where

Z

1 and

Z

2 are equal to

T

L

/ c

1 and

T

L

/ c

2

, respectively.


24


• Using Equations 1.36 and 1.39, the reflection ratio ( h

1 expressed as

/ g

1

) can be h

1 g

1

 

 



Z Z

1



Z Z

1



2

2

.

(1.40)

• The transmission ratio ( h

1

/ g

1

) can be written as g

2 g

1

 

 



2 Z

1

Z Z

1



2

.

(1.41)

• The ratio of the reflected wave and transmitted wave to the incident wave

T

L

/ c


25


• 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

Z

1 and the thick line has impedance

Z

2


26

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:



T

L cos

  

T

L

 dT

L

  d

   

L ds



2 x

 t

2

.

(1.42)

27



• The force and motion in the y direction can be written:



T

L sin

  

T

L

 dT

L

  d

   

L ds

 2 y

 t

2

, (1.43) where  expresses the slope of the string with respect to the arbitrary position of x

: tan

 

 y

 x

.

x axis at an

(1.44)

• The change of this slope with regard to a small change in written as x dx

  d

 



 y x







2 x

2 y dx

(1.45) using a Taylor expansion.

• Assuming that the displacement of the string is small enough to be linearized, then sin cos

 

1.

,

(1.46)

28



• Equations 1.42 and 1.43 thus become

 

L



T

L

 dT

L

  

L ds

 2 x

 t

2

,



T

L

 y

 x

 

T

L

 dT

L







 y



 

2

 x x y

2 dx





 

L ds

 2

 t

2 y

.

• The small ds can be rewritten as ds

 dy

2  dx 1



 

 y x

 2

 dx



1





1

2

 

 y x

 2



 .

(1.47)

(1.48)

(1.49)

• Its square can therefore be neglected compared to other variables.

Therefore, we can approximate ds

 dx .

(1.50)

• The small change of tension approximation as dT

L dT

L can be expressed by a first-order





T

L

 x dx .

(1.51)

29



• Equation 1.47 can be rewritten as



T

L

 x

 

L

 2 x

 t

2

.

• We can easily write Equation 1.48 as

T

L

 2 y

 x

2

 

L

 2 y

 t

2

.

• Rearranging Equation 1.53 results in





2 x y

2





T

L

L





2 t

2 y

.

• Equation 1.54 can be summarized as





2 x

2 y

 c

1 s

2





2 t

2 y

, where c s c s

2 

T

L



L

.

is the propagation speed of the string.


(1.55)

(1.56)

30

(1.52)

(1.53)

(1.54)


• Recall that the impedance of the string

Z is

Z =

T

L c s

.

• Using Equation 1.56, we can rewrite Equation 1.57 as

Z =

 c

L s

.

(1.57)

(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.


31

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, is propagation speed, the wavelength, f is the frequency in Hz (cycles/sec), and ω is the radian frequency in rad/sec) c s

λ is


32


• For mathematical convenience, we begin by expressing the waves in

Figure 1.13 using a complex function: y

   g

 x c t s



.

(1.59)

• The boundary condition at x



0 can be written as y

   g

  c t s

 

Y e



,

(1.60) where at x



Y e



0.

denotes the response of the string due to the excitation (

• We can therefore rewrite Equation 1.60 as

F e

 g

  c t s

 

Y e jk

  c t



,

(1.61) where we use the dispersion relation k

 

/ c s

.

• If we rearrange Equation 1.61 using an independent variable , then we obtain

 g

  

Y e

 jk



.

(1.62)

)

33



• We can therefore substitute α by g

  s

 

Y e



 s

 

, which gives us



Y e

 j

  t kx



.

• The velocity can be expressed using Equation 1.60:

(1.63) u y

 y

 x



0

  j



Y e



.

(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

= F e

  

T

L y

 x



0

.

(1.65)

• We can rewrite the impedance at the end as

Z m 0

 f y u y

 

 

  c

L s

.

(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.

34



• 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.

• The boundary condition at x=L requires that the displacement y ( x,t ) this boundary condition can be written as y

  always be 0. The solution that satisfies the governing wave equation and y

  

Y sin

   

.

(1.67)

• If we calculate the velocity using Equation 1.67 at x x =0 , then we have u y

 y

 x



0

   sin kL e



.

(1.68)

• The force at x=0

0 is f y

 

=



T

L y

 x



0



T kY cos kL e

L



.

(1.69)

35



• Equations 1.68 and 1.69 give us the impedance (specifically, the driving point impedance

Z m0

) at x



0

. That is,

Z m 0

 f y u y

 

 

 j

T

L c s cot kL

 j

 c

L s cot kL .

(1.70)

• When the wavelength is large compared to the length of the string, then

Equation 1.70 reduces to

Z m 0

 j

 c

L s

1 kL

.

(1.71)

• Rearranging this equation, we obtain

Z m 0

 j

T

L



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.

36



Figure 1.14 The driving point impedance of a finite string ( k is wave number and

L is the length of the string)


37


• Summary of Driving point impedances

Table 1.1 Driving point impedances

Nomenclature: ρ

λ

P

L

: Poisson’s ratio;

: mass per unit length of string, rod; ρ c s

: speed of propagation of string; c b

A



: mass per unit area of membrane;

Y /



; c p



Y

  

2 p

)

ρ : mass per volume of plate;

; 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;

T on frequency); v p bending moment of beam;

Z F m 0 m

: membrane tension (N/m); v

: propagation speed of plate (    p b

: propagation speed of bar ( =

, depending on frequency);

Z M m 0

: driving point impedance by shear force of beam and plate.

   b

, depending

: driving point impedance by


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

 2 dE

K



2



L dx , dE

P



T



L

   

2

  t 







1 dy dx T dx

2

L

 

 y x

 2

,

(1.73)

(1.74) where d expresses a small element.

39



• The total energy of the string can be written dE



1

2







L

 

 y t

 2



T

L

 

 y x

 2

 dx .

• Energy density can be expressed by

  dE

.

dx

• The total energy in the string can be written as

(1.75)

(1.76)

E

   dx



1

2

 c

L s

2



 





 y x





2







1

 y c s

 t

 2



 dx .

(1.77)

• Equation 1.77 demonstrates that the greater the slope along the string

(with regard to ) and the faster the speed of wave propagation, the more energy we have.

40



• Consider that we raise one end of the string (see Figure 1.16).

• The kinetic energy can be approximated as 1

2



L

 

2 c u

E

2 y

.

• The potential energy is 1

2

T

L



 u c

E y





2 c

E



; 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 ( time, and u y lifting velocity at the end)

T

L is tension along the string, c

E energy propagation speed, 


41


• These lead to the equation:

L u y

T u c

E y

 

1

2



L

  u

2 y



1

2

T

L



 u c

E y





2 c

E



, which gives us c

E

2 

T

L



L

.

(1.78)

(1.79)

• The speed of energy propagation is identical to the phase velocity of a string.


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.

43


1.2 From String Vibration to Wave

Sound Propagation

Chapter 1

Vibration and Waves

Yang-Hann Kim

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1.2 From String Vibration to Wave

Sound Propagation

Chapter 1

Vibration and Waves

Yang-Hann Kim

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