1. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Sound Propagation An Impedance Based Approach Vibration and Waves Yang-Hann Kim Chapter 1

2. 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

3. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.1 Introduction/Study Objectives Vibration can be considered as a special form of a wave (wave propagations, Figure 1.1). 3 Figure 1.1 The first, second, and third modes of a string (demonstration by C.-S. Park and S.-H. Lee, 2005, at KAIST)

4. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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 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 where Ф represents the phase difference between the second and third modes that are participating in the vibration. 4 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) (1.1)

5. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.2 From String Vibration to Wave There is also a phase difference in space, as demonstrated by Figure 1.3. 5 Figure 1.3 How the second and third modes create the vibration

6. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The first term of Equation 1.1 can be rewritten as Rearranging this equation in terms of x gives where L/(1 / f2) indicates a velocity that travels along the string. Equation 1.3 essentially means that there are two waves propagating along the string in opposite directions with a velocity of. Similarly, the first or even th mode can be interpreted in the same manner as for the second and third mode cases. 6 1.2 From String Vibration to Wave (1.2) (1.3)

7. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.2 From String Vibration to Wave The string vibration can generally be written as where Φ n is the phase of the n th mode. If we rewrite Equation 1.4 with respect to x, then This equation essentially states the following: “There are cosine waves propagating in the positive (+) and negative (−) directions with respect to space, x.” 7 (1.4) (1.5)

8. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.2 From String Vibration to Wave The general wave form, which is not simply a cosine wave, can be mathematically expressed as 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 (1.6)

9. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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. If we rewrite the function or wave g with regard to time, then we obtain Equation 1.7 states that the right-going wave in space can be seen as the wave propagates in time. 9 (1.7) Figure 1.4 The wave propagates in the positive (+) direction; g expresses the shape of the wave, c the wave propagation speed, and t and x are the time and coordinate

10. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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. 10 Figure 1.5 Wave diagram: waves can be observed at the x coordinate (space) and t axis (time), where Δt denotes infinitesimal time, and indicates arbitrary position and time, and y is the wave amplitude

11. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The sine wave is a special wave that can be expressed by Equation 1.7. The sine wave, propagating to the right, is expressed 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 where. It relates the variable that expresses the changes of space (x), k, with that related to time (t), ω. That is, 11 1.2 From String Vibration to Wave (1.8) (1.9) (1.10)

12. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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 We can rewrite Equation 1.11 as where k represents the number of waves per unit length ( ). We call this the wave number or a propagation constant. 12 (1.12) (1.11)

13. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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). 13 1.2 From String Vibration to Wave 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

14. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.2 From String Vibration to Wave We can also obtain an additional relation from Equations 1.10 and 1.12. That is, This states that the variables which express space (λ) and time (f ) are not independent of each other.  “dispersion relation” By using a complex function, Equation 1.9 can be rewritten as where Y is the complex amplitude. For the sake of simplicity, Equation 1.14 will be written as We can also express Equation 1.15 with respect to time instead of space, that is 14 (1.13) (1.14) (1.15) (1.16)

15. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.3 One-dimensional Wave Equation Any one-dimensional wave can be expressed as 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: where denotes the derivative of each function with respect to its arguments (e.g., ). Its time rate of change is expressed as which leads to 15 (1.17) (1.18) (1.19) (1.20) (1.21)

16. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.3 One-dimensional Wave Equation 16 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) Figure 1.7 illustrates the associated kinematics of the right-going and left- going wave.

17. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.3 One-dimensional Wave Equation 17 If we differentiate Equations 1.20 and 1.21, we obtain Any one-dimensional wave ( ) which has left-going and right-going waves with respect to the selected coordinates satisfies the partial differential equation: Equation 1.23 can then be rewritten as (1.22) (1.23) (1.24)

18. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.3 One-dimensional Wave Equation A three-dimensional version of Equation 1.24 can be written as where denotes the amplitude of three-dimensional wave. 18 (1.25) The boundary condition can generally be written as 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. (1.26)

19. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.3 One-dimensional Wave Equation Two types of boundary conditions: passive and active 19 Figure 1.8 Examples of passive and active boundary conditions

20. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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. We wish to determine the relation between the incident wave g 1, the reflected wave h 1 and the transmitted wave g 2. 20 Figure 1.9 Waves in two strings of different thickness ( g 1 is an incident wave, h 1 represents a reflected wave, and g 2 is a transmitted wave)

21. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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 ( 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 Denote the waves on the negative x axis region, #1 string, as y 1 and express the wave that propagates in the positive x axis as y 2. Describing these waves with regard to time, they can be written as 21 (1.27) (1.28) (1.29) (1.30)

22. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The velocity in the y direction at can be written as At, it is We therefore obtain the following equality since the velocity must be continuous: 22 1.4 Specific Impedance (Reflection & Transmission) (1.31) (1.32) (1.33) The forces in the y direction ( ) are related to the tension along the string a and the slope (Figure 1.10) as Therefore, we can rewrite Equation 1.28 as (1.34) (1.35)

23. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.4 Specific Impedance (Reflection & Transmission) 23 Figure 1.10 Forces acting on the end of the string where T L is tension, f y describes the force in the y direction, y indicates the amplitude of the string, and x denotes the coordinate)

24. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.4 Specific Impedance (Reflection & Transmission) We can postulate that the string’s wave amplitude at is zero. We can therefore write Equations 1.33 and 1.35 as The ratio of the string’s force in the y direction ( f y ) and the associated velocity ( u y ) can be written as 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 where Z 1 and Z 2 are equal to and, respectively. 24 (1.36) (1.37) (1.38) (1.39)

25. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Using Equations 1.36 and 1.39, the reflection ratio ( h 1 /g 1 ) can be expressed as The transmission ratio ( h 1 /g 1 ) can be written as The ratio of the reflected wave and transmitted wave to the incident wave depends entirely on the string’s impedance,. 25 1.4 Specific Impedance (Reflection & Transmission) (1.40) (1.41)

26. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 26 1.4 Specific Impedance (Reflection & Transmission) 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 and the thick line has impedance 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.

27. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.5 The Governing Equation of a String Let us examine an infinitesimal element of string (Figure 1.12). Newton’s second law in the x direction can be written: 27 Figure 1.12 Newton’s second law on an infinitesimal element of a string (notation as for Figure 1.10) (1.42)

28. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.5 The Governing Equation of a String The force and motion in the y direction can be written: where expresses the slope of the string with respect to the x axis at an arbitrary position of x : The change of this slope with regard to a small change in x ( ) can be written as using a Taylor expansion. Assuming that the displacement of the string is small enough to be linearized, then 28 (1.44) (1.45) (1.46) (1.43)

29. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.5 The Governing Equation of a String Equations 1.42 and 1.43 thus become The small can be rewritten as Its square can therefore be neglected compared to other variables. Therefore, we can approximate The small change of tension can be expressed by a first-order approximation as 29 (1.47) (1.48) (1.49) (1.50) (1.51)

30. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.5 The Governing Equation of a String Equation 1.47 can be rewritten as We can easily write Equation 1.48 as Rearranging Equation 1.53 results in Equation 1.54 can be summarized as where c s is the propagation speed of the string. 30 (1.52) (1.53) (1.54) (1.55) (1.56)

31. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.5 The Governing Equation of a String Recall that the impedance of the string is Using Equation 1.56, we can rewrite Equation 1.57 as 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 (1.57) (1.58)

32. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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. 32 Figure 1.13 Wave propagation by harmonically exciting one end of a semi-infinite string ( T is period, is propagation speed, λ is the wavelength, f is the frequency in Hz (cycles/sec), and ω is the radian frequency in rad/sec)

33. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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: The boundary condition at can be written as where denotes the response of the string due to the excitation ( ) at We can therefore rewrite Equation 1.60 as where we use the dispersion relation. If we rearrange Equation 1.61 using an independent variable, then we obtain 33 (1.59) (1.60) (1.61) (1.62)

34. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd We can therefore substitute α by, which gives us The velocity can be expressed using Equation 1.60: The force at the end of the string is related to the tension and the slope of string (Figure 1.10): We can rewrite the impedance at the end as 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 1.6 Forced Response of a String: Driving Point Impedance (1.64) (1.65) (1.66) (1.63)

35. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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. The boundary condition at x=L requires that the displacement y(x,t) always be 0. The solution that satisfies the governing wave equation and this boundary condition can be written as If we calculate the velocity using Equation 1.67 at x =0, then we have The force at x=0 is 35 (1.67) (1.68) (1.69)

36. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equations 1.68 and 1.69 give us the impedance (specifically, the driving point impedance Z m0 ) at. That is, When the wavelength is large compared to the length of the string, then Equation 1.70 reduces to 36 1.6 Forced Response of a String: Driving Point Impedance (1.70) (1.71) Rearranging this equation, we obtain 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. (1.72)

37. 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 Figure 1.14 The driving point impedance of a finite string ( k is wave number and L is the length of the string)

38. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Summary of Driving point impedances 38 1.6 Forced Response of a String: Driving Point Impedance Nomenclature: ρ L : mass per unit length of string, rod; ρ A : mass per unit area of membrane; ρ : mass per volume of plate; λ P : Poisson’s ratio; c s : speed of propagation of string; ; 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 m : membrane tension (N/m); v b : propagation speed of bar (=, depending on frequency); v p : propagation speed of plate (aaaaaaa, depending on frequency); : driving point impedance by bending moment of beam; : driving point impedance by shear force of beam and plate. Table 1.1 Driving point impedances

39. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.7 Wave Energy Propagation along a String Let’s determine how much energy can be stored in an infinitesimal element of string. The kinetic and potential energy in the infinitesimal element of the string can be written where d expresses a small element. 39 Figure 1.15 The change of an infinitesimal element of a string in infinitesimal time (1.73) (1.74)

40. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The total energy of the string can be written Energy density can be expressed by The total energy in the string can be written as 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 1.7 Wave Energy Propagation along a String (1.75) (1.76) (1.77)

41. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Consider that we raise one end of the string (see Figure 1.16). The kinetic energy can be approximated as. The potential energy is ; this can be readily obtained by the work done due to the elongation of string. 41 1.7 Wave Energy Propagation along a String Figure 1.16 Energy propagates along a string by raising one end ( is tension along the string, energy propagation speed, time, and lifting velocity at the end)

42. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 1.7 Wave Energy Propagation along a String 42 These lead to the equation: which gives us The speed of energy propagation is identical to the phase velocity of a string. (1.78) (1.79)

43. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 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