1. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Sound Propagation An Impedance Based Approach Radiation, Scattering, and Diffraction Yang-Hann Kim Chapter 4

2. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Outline 4.1 Introduction/Study Objectives 4.2 Radiation of a Breathing Sphere and a Trembling Sphere 4.3 Radiation from a Baffled Piston 4.4 Radiation from a Finite Vibrating Plate 4.5 Diffraction and Scattering 4.6 Chapter Summary 4.7 Essentials of Radiation, Scattering, and Diffraction 2

3. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.1 Introduction/Study Objectives Scattering and diffraction are physical phenomena, which represent waves deflected by characteristics of the discontinuity. Both can therefore be expressed by solutions of the wave equation which satisfy boundary conditions. 3 Figure 4.1 (a) Radiation, (b) Scattering, and (c) Diffraction: waves are visualized using a ripple tank. Incidence waves for (b)-(d) are plane waves coming from the left. The depth of the water has to be sufficiently smaller than 1/8 th the wavelength to create a non-dispersive wave( has to be smaller than roughly 0.5, where is wave number, and h is water depth)

4. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.2 Radiation of a Breathing Sphere and a Trembling Sphere To understand these rather complicated phenomena, we need to understand sound fields induced by the basic unit sources: the radiation of a breathing sphere and a trembling sphere. The first type of basic radiation unit is a breathing sphere. 4 Figure 4.2 A breathing sphere and its radiation pattern: is the radius, indicates the radial distance, and denotes the velocity magnitude

5. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Linear differential equation in polar coordinates can be expressed as The solution of Equation 4.1 can be written as 5 4.2 Radiation of a Breathing Sphere and a Trembling Sphere (4.1) (4.2) where is the wave number in the direction. If the surface of the sphere harmonically vibrates, then the velocity on the surface ( ) can be written as (4.3) where is the velocity potential.

6. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The rate of change of the velocity potential with regard to r has to be the velocity at the sphere (4.3). That is, The velocity at will then be 6 (4.4) (4.5) (4.6) From Equations 4.3 and 4.5, we can obtain

7. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The velocity potential, the velocity, and the acoustic pressure can therefore be written as Note that Equations 4.7, 4.8, 4.9 are only valid if r is larger than a. Note that the velocity and pressure depend on the relative scales such as the ratio between the radius of the sphere ( ) and the observation position ( ): the sphere’s radius with respect to the wavelength ( ) and the observation position with regard to the wavelength ( ). 7 4.2 Radiation of a Breathing Sphere and a Trembling Sphere (4.7) (4.8) (4.9)

8. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The acoustic impedance of the sphere ( ) can be obtained from Equations 4.8 and 4.9, that is, 8 4.2 Radiation of a Breathing Sphere and a Trembling Sphere “Far-field” : when is much larger than 1  The radiation from the breathing sphere resembles a one-dimensional acoustic wave (plane wave), because the acoustic impedance is. “Near-field” : when is smaller than 1  The imaginary part (i.e., the reactive part of the impedance) dominates the radiation characteristics. Therefore, the radiation is not likely to be effective. (4.10)

9. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The radiation impedance of a breathing sphere at r=a exhibits how well the sphere radiates sound from its surface. This can be obtained from Equation 4.10, that is, The mechanical impedance ( ) can be expressed by multiplying Equation 4.11 by the surface area ( ), that is, 9 4.2 Radiation of a Breathing Sphere and a Trembling Sphere (4.11) (4.12)

10. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 10 Figure 4.3 Impedances of the breathing sphere: (a) the acoustic impedance and the radiation impedance, and (b) the radiation power. ( and dominate the characteristics of the impedances; as they become larger, the wave behaves as if it is planar) 4.2 Radiation of a Breathing Sphere and a Trembling Sphere

11. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd If we calculate the mean intensity ( ) by using Equations 4.8 and 4.9, then we obtain 11 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The radiation power can be obtained from Equation 4.17 by multiplying the area of interest by (4.17) (4.18) where P and U are the sound pressure magnitude (4.9) and the complex conjugate of the velocity magnitude (4.8).

12. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 12 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The normalized radiation power ( ) of a circular plate of radius a with a velocity of U0 and frequency w. This is expressed as (4.19) which highlights that the radiation power becomes larger as we increase ka (Figure 4.3(b)).

13. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The second type of basic radiation unit is a trembling sphere (Figure 4.4). 13 4.2 Radiation of a Breathing Sphere and a Trembling Sphere Figure 4.4 The trembling sphere. (The direction of vibration is z, the velocity magnitude is, and Ur is the velocity in the r direction. Other symbols represent coordinates)

14. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The acoustic waves generated by this trembling sphere would satisfy the acoustic wave equation in the spherical coordinate, that is, 14 4.2 Radiation of a Breathing Sphere and a Trembling Sphere (4.20) (4.21) The boundary condition on the surface of the trembling sphere ( r=a ) can be written as where Ur=a denotes the velocity of the trembling sphere in the r direction. The solution satisfying Equation 4.20 is found to be (4.22)

15. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 15 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The pressure (p ) can be calculated from Equation 4.22 as (4.23) The velocity ( ) can be obtained by taking the derivative of the potential function with respect to r as (4.24) Equations 4.21 and 4.24 lead us to write and we can obtain as (4.25) (4.26)

16. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The pressure magnitude ( ) can be obtained by using Equations 4.23 and 4.26, that is, 16 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The velocity ( ) can be calculated using Equations 4.24 and 4.26, that is (4.28) (4.27) The acoustic impedance ( Z) can be obtained from Equations 4.27 and 4.28 as (4.29) which states that the impedance is independent of θ although the pressure and velocity magnitude depend on cosθ.

17. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 17 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The radiation impedance ( ) can be obtained from Equation 4.29, that is, (4.30) If ka is small (i.e., if the radius of the sphere is small compared to the wavelength of interest), then the reactive term which is the imaginary part of the impedance dominates the radiation characteristics. When ka is large, then the resistive term governs the impedance and the trembling sphere effectively radiates sound waves.

18. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 18 4.2 Radiation of a Breathing Sphere and a Trembling Sphere The normalized radiation power ( ) can be expressed by (4.31) To effectively express this angle dependency of the radiation, we define the directivity factor as (4.32) where Ispheis the intensity radiated from the breathing sphere with radiation power equivalent to that of the radiator of interest. denotes the intensity of the radiator whose directivity we wish to characterize. Note that the pressure and velocity depend on θ.

19. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 19 4.2 Radiation of a Breathing Sphere and a Trembling Sphere Figure 4.5 The directivity factor of a trembling sphere; the circumferential angle is expressed in degrees and the radial distance is a non-dimensional arbitrary unit

20. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 20 4.2 Radiation of a Breathing Sphere and a Trembling Sphere Figure 4.6 Impedances of the trembling sphere: (a) the acoustic impedance and the radiation impedance, and (b) the radiation power. Note that the radiation impedance and the radiation power are proportional to and, respectively. The corresponding characteristics of the breathing sphere are proportional to and. The radiation of the trembling sphere therefore depends on the viewing location and the size of the diameter relative to the wavelength of interest

21. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston 21 Figure 4.7 Volume integral of the Kirchhoff-Helmholtz integral equation. (Sp and Suexpress the boundary surface for the pressure and velocity U. d denotes a surface that is infinitely far from the origin, and and indicate the observation position and the boundary position vector, respectively) We can rewrite the integral equation that emphasizes the individual contribution of pressure and velocity sources as (4.33)

22. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston Consider the sound radiation from a baffled piston as illustrated in Figure 4.8. Note that we only have the second integral of Equation 4.33 in this case. 22 Figure 4.8 Surface integral to calculate the sound radiation from a baffled piston and nomenclature of the coordinate

23. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston 23 If we apply Equation 4.33 to this specific case by employing the surface of the integral ( ) (Figure 4.8), then we obtain (4.34) Note that Equation 4.34 states that the radiated sound pressure is induced by the vibrating surface velocity. Equation 4.34 can also be regarded as an expression of Huygen’s principle in an integral form.

24. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston We start with the case where Un is constant. In other words, the piston is a rigid vibrator (Figure 4.9). By using Equation 4.34, the pressure at an arbitrary position ( z ) can be expressed by 24 Figure 4.9 The radiated sound field from an infinitely baffled circular piston (4.35)

25. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston Equation 4.35 can be simplified as 25 where is the distance from the edge of a circular piston to z (detailed derivation is given in Section 4.7.2). (4.36) There are two distinct contributions in the z direction. The first term ( ) is the pressure coming from the center of the piston, and the second term ( ) is from the piston’s rim. The coefficient ( ) is the pressure of the plane wave that is generated by the piston’s motion. Note that these two pressure waves interfere with each other. The interference sometimes mutually cancels the waves or reinforces them. If z is significantly large compared to the radius of the piston, then Equation 4.36 becomes (4.37)

26. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston Figure 4.10 depicts the coordinates and nomenclature that we use to predict the radiation from the vibrating piston to an arbitrary position, including the z axis. The sound pressure ( ) on the plane is given as 26 Figure 4.10 Coordinate set-up and variables for obtaining radiated sound field on x-z plane from an infinitely baffled circular piston (4.38) where is a Bessel function of the first kind.

27. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston It is interesting to look at the sound pressure at, that is, 27 (4.39) This is identical to Equation 4.37, meaning that Equation 4.38 is a general expression of the radiation sound pressure on the x -z plane. Using Equations 4.38 and 4.39, we can obtain an expression which provides further significant physical insight, that is, (4.40) Also, note that the mean intensity is (4.41)

28. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston Therefore, the intensity ratio between the time-averaged intensity at an arbitrary position with respect to that of the axis attached to the center of the piston can be expressed as 28 (4.42) Equation 4.42 is often called the directivity or directivity index. Other names such as spreading and spreading index are also widely used. The angular dependency of the radiation significantly decreases as the radiator size decreases relative to the wavelength, or when we have lower frequency radiation. On the other hand, when ka becomes larger, or the frequency becomes higher, the radiation strongly depends on the angle. by using Equations 4.40 and 4.41.

29. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston 29 Figure 4.11 The directivity or directivity index of a baffled circular piston

30. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston Note that the radiation impedance of the plate varies with the position of the plate. The velocity is constant (Un) on the surface, but the resulting pressure is not uniform. The radiation impedance of this case can be defined as 30 (4.43) where F is force acting on the source surface (S), and is the average pressure on the surface. To obtain F, the pressure on the surface of the piston is needed. This can be regarded as the sum of the pressure induced by the other area over the entire piston surface, that is, (4.44) where Figure 4.12(a) illustrates the variables for the integration.

31. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston 31 The force acting on the piston surface can be given by (4.45) The integration of Equation 4.45 can be written as: (4.46) where is the first kind of Bessel function of zero order, and is the zero-order Struve function.

32. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston 32 The force acting on the piston surface can then be given by (4.47) where aaaaa is the first kind of Bessel function of the first order, and aaaaaais the Struve function of the first order. Figure 4.12 The variables on the surface of the disk for integrating with respect to (a) z and f, and (b) R and y (b) (a)

33. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.3 Radiation from a Baffled Piston 33 is then defined as (4.48) and the radiation impedance of a baffled circular piston with radius a is (4.49) Figure 4.13 Radiation impedance, which is normalized to, of the circular baffled piston with respect to. The solid line represents the resistance term, and the dashed line the reactance

34. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate The radiation due to the plate vibration can be considered to be composed of the radiation of n modes of vibration. The radiation due to each mode of vibration can be superimposed by the vibration of many pistons, as illustrated in Figure 4.14(b). We can assume that each mode of vibration can have equivalent pistons. 34 Figure 4.14 (a) Radiation from a finite plate and (b) its possible modeling (b) (a)

35. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate To understand the radiation from the vibrating plate, we look at basic radiations patterns such as those illustrated in Figure 4.15. The radiated fields are obtained using the Rayleigh integral equation. The radiator’s typical dimensions are larger than the wavelengths that are generated; the radiation efficiencies are therefore fairly good. The sound pressures at the distance from the radiators ( ) become negligible when the distance is long compared to the wavelength, except for case (a). This is because the far field sound propagation tends to a plane wave, resulting in a perfect cancellation for the cases of (b) and (c) in Figure 4.15. This kind of cancellation becomes more and more significant when we have higher order modes. 35

36. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate 36 Figure 4.15 Basic examples of plate vibration and radiation ( A indicates the observation position, and can be expressed as ( 0,0,z l ) in the Cartesian coordinate)

37. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate To look more specifically at the radiation from a plate, we describe the waves in space using rectangular coordinates for convenience. The wave is assumed to be harmonic in space and time, without loss of generality. The acoustic wave can now be written as 37 Equation 4.50 must satisfy the linear acoustic wave equation; the following equality must therefore hold, that is, (4.50) (4.51) (4.52) (4.53) where and are the length of the plate in the x and y directions, and m and n are integers. where, which is a dispersion relation. The wave numbers in the x and y directions ( and ) can be written as

38. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate Figure 4.16(a), Equations 4.51, 4.52, and 4.53 tell us that the wave number in the z direction ( ), which describes how the wave in the z direction propagates, could be real or imaginary. 38 Figure 4.16 Radiation from a finite plate: (a) corner mode, edge modes, and radiation circle in wave number domain and (b) edge (corner) mode dominates the radiation in the left- (right-) hand image

39. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate The development of either exponentially decaying propagation (evanescent wave) in the z direction or a continuous phase changing wave is determined by the location (ox, ky) in reference to the radiation circle. 39 (4.54) (4.55) Case 2) When is located outside a radiation circle, has to be imaginary and the wave therefore decays exponentially. The wave is less likely to propagate in the z direction with a larger or, in other words, as the wavelength becomes increasingly smaller. Case 1) When is located inside a radiation circle, The propagator can be mathematically written as. The wave in the z direction continuously changes its phase as it propagates.

40. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate In acoustic holography, the spatial distribution of a sound wave is expressed in terms of the wave number domain of interest, and then propagated to a plane that is not measured by using a propagator. Figures 4.17 and 4.18 illustrate the basic procedures of acoustic holography. 40 Figure 4.17 Conceptual diagram of acoustic holography in rectangular coordinate

41. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.4 Radiation from a Finite Vibrating Plate 41 Figure 4.18 Illustration of acoustic holography and its fundamental procedure

42. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering Figure 4.19 illustrates a typical diffraction and scattering phenomena. The diffraction tends to be stronger as the wavelength becomes larger. In other words, we have more diffraction at the back of the wall as the wavelength increases. Diffraction is generally used to describe the physical circumstances under which we can hear sound but cannot see the sound source. Scattering describes waves that induced due to an abrupt impedance change in space, especially when the waves spread out in space. 42 Figure 4.19 Diffraction around a straight barrier (note that the diffraction strongly depends on the wavelength)

43. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering Beginning with the simplest case which considers all the necessary fundamentals, suppose that we have a plane wave impinging on an arbitrary scatterer, as illustrated in Figure 4.20. 43 (4.58) The total sound pressure can be expressed as (b)(a) Figure 4.20 Scattering by (a) arbitrary scatterer and by (b) a sphere. (P i, Pscrepresent the complex amplitude of incident and scattered waves;0 and n represent the surface of the scatterer and a unit vector normal to the surface, and a is the characteristic length of the scatterer)

44. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering The total pressure has to satisfy the following boundary condition, that is, 44 (4.59) The boundary is acoustically rigid and is a unit normal vector on the surface. Let P propagate in the direction of the wave number vector k. The incident wave at the position r can then be written as (4.60) where B is the complex amplitude of the incident wave (Figure 4.20). Equations 4.58, 4,59 and 4.60 lead us to a relation between the scattered wave and the incident wave, that is, (4.61)

45. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering 45 If we rewrite Equation 4.61 in terms of the velocity of the scatterer by using the linearized Euler equation, then we obtain (4.62) (4.63) where we assume that the scattering velocity ( ) is harmonic in time, that is, where is the magnitude of the scattering velocity in vector form. By starting with the simplest case, we consider the scattering of a rigid sphere to explore what is meant by Equations 4.61 and 4.62. Rewriting Equation 4.61 using the coordinate of Figure 4.20(b), we obtain (4.64) where represents the angle between the propagation vector and the normal vector.

46. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering 46 The velocity in the normal direction, which is, that is, where is the surface of the sphere (Figure 4.20(b)). Since the radius of the sphere is a, we can rewrite Equation 4.65 as (4.66) (4.65) (4.67) If the size of the scatterer is substantially smaller than the wavelength of interest (i.e., if ), then Equation 4.66 can be approximately rewritten as This means that the scattering field is essentially induced by the radiation of the sphere, which vibrates with a velocity as described by Equation 4.67.

47. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering 47 Figure 4.21 Directional component ( ) of the normal velocity ( ) on the rigid sphere with respect to ka

48. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering 48 Figure 4.22 (a) Two-dimensional and (b) three-dimensional rectangular slit with corresponding nomenclature (b)(a) Figure 4.22 exhibits two fundamental scatterers which can demonstrate the scatterers’ geometrical effect on the radiation. The first is a two- dimensional case and the second is its extension to three dimensions.

49. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering The radiated sound pressure for the two-dimensional slit can be found as 49 (4.70) The scattered field of the rectangular slit can be obtained as (4.71) We can also consider that Equations 4.70 and 4.71 describing a diffraction field due to the slits or scatterers. It can also be argued that the diffractions in these cases strongly depend on the non-dimensional scale factors (ka and kb) and observation angle. (see Section 4.7.4.5 for details). (See Section 4.7.4.5 for the detailed derivation.)

50. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering To understand the diffraction phenomenon, we begin by studying a typical example: diffraction phenomenon by a sound barrier as shown in Figure 4.23, for example. 50 Figure 4.23 Two-dimensional diffraction problem for a plane wave source (it is assumed that the wavelength is much larger than the thickness of the wall, and the wall is acoustically rigid): (a) semi-infinite and (b) finite barrier case (b) (a)

51. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering The closed form solution that describes diffraction due to a semi-infinite barrier (Figure 4.23(a)) can be obtained (see Section 4.7.6.3): 51 where (4.72) (4.73) We can say that the diffraction is the result of the edge scattering. In other words, we hear sound coming from the sound sources. This means that the edge condition strongly affects the diffraction in the shadow zone. When, we cannot see the sound source. In other words, we are in the shadow zone and the diffraction is dominated by the scattered field induced at the edge of the wall. When, we can see the sound source on the positive z axis. The diffraction field is composed of two parts: the scattered field from the edge of the wall and the direct sound field.

52. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering 52 Figure 4.24 Diffraction of barriers when we have a monopole source, obtained using FDTD (Finite Difference Time Domain): (a) by a straight barrier and (b) by a curved barrier with respect to time ( S denotes a monopole source on the ground). (Photographs courtesy of H. Tachiban, University of Tokyo.)

53. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering Figures 4.25 and 4.26 effectively provide the practical parameters that are associated with the diffraction of a barrier. As the wavelength becomes increasingly larger (much larger than the distance A+B), the barrier’s height would lose its presence in terms of diffraction. On the other hand, if the wavelength is much smaller than A+B, then the listener would perceive that the sound comes from the edge of the barrier. 53 Figure 4.25 The sound barrier and associated nomenclature, where NF is the Fresnel number, S represents the source position and R is the receiver’s location. We also assume that the wall thickness is small relative to the wavelength, and acoustically hard

54. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering The Fresnel number is defined 54 (4.74) The transmission loss of a barrier ( ) is generally expressed as (4.75) Figure 4.26 The sound attenuation due to a sound barrier (NF is the Fresnel number)

55. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering Refraction is generally due to a change in the media’s characteristic impedance. For example, as illustrated in Figure 4.27, we can observe the refraction of sound due to the inhomogeneous characteristics of the media because of varying temperature. 55 Figure 4.27 Refraction due to the media’s impedance change ( T is temperature and H is height) (a) during the day and (b) during the night

56. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.5 Diffraction and Scattering To understand this in more depth, consider the case of multi-layer media as illustrated in Figure 4.28. As the propagation vectors depict, the propagation becomes stiffer in the x direction as we have greater propagation speed in the y direction. 56 Figure 4.28 Sound propagation in a medium where the characteristic impedance changes smoothly

57. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 4.6 Summary The breathing sphere and the trembling sphere are basic units that can create any radiation field by their linear combinations. The radiations are mainly dominated by the relative size (ka) of the radiator compared to the wavelength, and the radiated sound field is mainly governed by the relative distance (kr ) from the radiator compared to the wavelength of interest. The scattered sound field depends on - the angle of the incident wave - the boundary condition of a scatterer - the ratio of the wavelength relative to the size of a scatterer Diffraction also depends on - the angle of the incident wave - the angle from the edge The Fresnel number is widely accepted as a practical means to design a barrier. 57