1. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Sound Propagation An Impedance Based Approach Acoustics in a Closed Space Yang-Hann Kim Chapter 5

2. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Outline 5.1 Introduction/Study Objectives 5.2 Acoustic Characteristics of a Closed Space 5.3 Theory for Acoustically Large Space (Sabine’s Theory) 5.4 Direct and Reverberant Field 5.5 Analysis Methods for a Closed Space 5.6 Characteristics of Sound in a Small Space 5.7 Duct Acoustics 5.8 Chapter Summary 5.9 Essentials of Acoustics in a Closed Space 2

3. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.1 Introduction/Study Objectives Depending on the distribution of the impedance, the sound propagation differs significantly. Sound propagation will be determined by the overall volume of the space and the wall impedances which characterize the space. The volume of space has to be considered with regard to the wavelength of interest. – If the volume is fairly large, the waves would behave as if in a large space, and would reach all possible places. – If the volume is small compared to the wavelength, then the wave would appear to be everywhere in the space instantly. 3

4. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.2 Acoustic Characteristics of a Closed Space It is usually not plausible to express the sound that is likely to propagate in a space of interest mathematically. The volume of a space of interest determines the major acoustical characteristics of sound propagation in the space. Intuitively, a measure has to be scaled with respect to the wavelength of interest. For an acoustically large space, Sabine found that the reverberation period represents the acoustic characteristics of the space well. 4 Acoustically small spaceThe fluid particles in the space can be regarded as if they are all moving with the same phase. Acoustically large space The acoustic wave travels in the space as a ray. : wavelength : representative length of the space, where V is the volume of the space

5. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) The spatial distribution of the acoustic waves is not well dependent upon the location of the space. In other words, if the pressure is measured at any position in the space, it would be almost identical to the mean value. This phenomenon would be more likely if more randomly distributed wall impedance exists. A diffuse field implies a space in which the sound is likely to be equally distributed irrespective of the position. 5 Figure 5.1 Illustration of sound propagation patterns in rooms. (These are the cases in which the size of room is acoustically large; in other words, the typical dimension of the room is much larger than the typical wavelength)

6. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) We first define acoustic energy density as The sound energy at an arbitrary location is not expected to be perfectly uniform. If considering an averaged sound energy density with respect to a certain time and a small volume, expressed as If a diffuse field is expressed using this measure, then the sound field would satisfy the equality. 6 (5.1) kinetic energy per unit volumepotential energy induced by the expansion and contraction of the unit volume of the medium (5.2)

7. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) 7 The sound field before the sound wave is reflected from the walls (direct sound field) is quite different compared to after the wave has been reflected as sound from the walls (reverberant sound field or reflected sound field). The sound energy of a reverberant field can be determined using an equation that expresses the conservation of sound energy (Equation 2.36): Figure 5.1 Illustration of sound propagation patterns in rooms. (These are the cases in which the size of room is acoustically large; in other words, the typical dimension of the room is much larger than the typical wavelength) (5.3)

8. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) 8 Figure 5.2 The energy balance by the power through the boundary ( V is volume, П in is the incoming power, and П out is the outgoing power) With the assumption that the volume does not include any sound source bounded by the surface of the room as well as by the sound source, if Equation 5.3 is integrated with regard to the volume, we have (5.4) ( : power)

9. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) It is possible to regard the sound in a closed space as being composed by two sound fields: the first is direct and the second is reverberant. If Equation 5.5 is applied when only a reverberant sound field exists, the energy conservation equation for the reverberant sound is: 9 (5.5) (5.6) loss induced by the direct sound loss induced by the reverberant sounds

10. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) Sabine found that the reverberant sound field created by the reflection from the walls can be regarded as a diffuse sound field. Equation 5.6 can be rewritten as Sabine also noted that To convert Equation 5.8 into a formula, a coefficient that has a time scale must be used. Here, time scale is denoted as τ. Equation 5.7 and 5.8 then lead to 10 (5.7) (5.8) (5.9) (5.10)

11. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) The concept of energy decay as expressed by 1/ τ or the characteristic decay time ( τ ) is strongly related to the walls that form a closed space as well as the items located in the space, as these items act as sound absorbing elements. They can be regarded as an open window that dissipates sound energy from the closed space to outside.  concept of the “area of an open window” 11 (5.11) area of the open window

12. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) Intuitively, it is natural to postulate that a greater size would lead to a longer time required to dissipate the acoustic energy in the room. Equation 5.11 could be rewritten in the proportional form: Sabine successfully found a coefficient that can convert the proportional form of Equation 5.12 into the following equality: This equation essentially states that the sound in the room (strictly speaking, the sound in a diffuse field) can be represented by only one parameter: the characteristic decay time τ. 12 (5.12) (5.13) ( c : speed of sound)

13. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) T 60 (the reverberation time or the reverberation period) is defined as the time required to reduce the sound by 60dB. Applying this definition to Equation 5.10 yields Rearranging Equation 5.14 provides Equations 5.13 and 5.15 result in where the area of the open window A s can be rewritten as where N is the number of elements that comprise the room of interest, α n is the absorption coefficient (which is the ratio of the absorbed sound power to the incident sound power), and n is an index that represents each material. 13 (5.14) (5.15) (5.16) (5.17)

14. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) An expression that relates the reverberation period to the open window area and the volume of the closed space is found to be 14 (5.18) Table 5.1 Reverberation time of famous concert halls (RT oc and RT unoc are reverberation time when occupied and unoccupied) (Adapted from L.L. Beranek, Concert Halls and Opera Houses: Music, Acoustics, and Architecture. Springer-Verlag, New York Inc., © 2004.)

15. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.3 Theory for Acoustically Large Space (Sabine’s theory) 15 Figure 5.3 Several famous concert halls: (a) Vienna Grosser Musikvereinsaal ( Concert Halls and Opera Houses, 2nd edition, 2004, pp.174, ‘‘Vienna Grosser Musikvereinssaal,’’ L. Beranek, Springer-Verlag New York, Inc.: With kind permission of Springer Science + Business Media); (b) Berlin Philharmonie ( Concert Halls and Opera Houses, 2nd edition, 2004, pp. 298, “Berlin Philharmonie,” L. Beranek, Springer-Verlag New York, Inc.: With kind permission of Springer Science + Business Media.); (c) Tokyo Suntory Hall ( Concert Halls and Opera Houses, 2nd edition, 2004, pp.408, ‘‘Tokyo Suntory Hall,’’ L. Beranek, Springer- Verlag New York, Inc.: With kind permission of Springer Science + Business Media.); and (d) Boston Symphony Hall ( Concert Halls and Opera Houses, 2nd edition, 2004, pp.48, ‘‘Boston Symphony Hall,’’ L.L. Beranek, Springer-Verlag New York, Inc.: With kind permission of Springer Science + Business Media.) (a) (b) (c) (d)

16. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field A direct sound field refers to a field that does not have any reflected sound waves. If there is no reflection, then the total sound power through the surface at r 1 or r 2 has to be conserved provided that there are no energy loss in the medium. 16 Figure 5.4 Sound propagation in an open space (direct sound field)

17. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field If r 2 = 2r 1, then the intensity ratio is This indicates that the sound intensity will be reduced by 6dB. For a diffuse field, the acoustic properties are independent with respect to the location. The solution of Equation 5.21 is therefore 0dB. 17 (5.19) ( : magnitude of intensity at r 1 or r 2 ) (5.20) (5.21)

18. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field The sound that we hear is generally the sum of the direct and the reverberant sound. The direct sound would be dominant if a listener is close to the source; however, reverberant sound would be more likely to dominate when the listener is further away from the sources and close to the wall or walls. 18 Figure 5.5 Spatial variation of sound field with respect to the distance from source

19. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field It is necessary to derive a certain measure or scale that can determine the degree of participation of the direct and reverberant fields, or the direct and reflected sound waves in a room. For a steady state condition, Equation 5.5 can be rewritten as The sound power generated by the sound sources is balanced by the sound power reflected due to the direct sound and due to what is induced by the reverberant sound on the surface that we select. How much is reflected is directly related to the absorption coefficient of the walls. The average absorption coefficient of the walls is denoted 19 (5.22) (5.23) A t : total area of the closed space A s : equivalent area of an open window

20. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field П out, direct, which is the power reflected from the walls by the incident sound power ( П in, direct ), are related as Equations 5.22 and 5.24, the time rate change of the reverberant sound energy, are related to the direct sound power, that is: The sound power passing through the surface of a sphere with a radius of r has to be identical to what the sound source generates. This physical balance can be mathematically written as 20 (5.24) (5.25) (5.26) ( I r : intensity at distance r from the source)

21. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field If a monopole source and the radius are widely spaced relative to the wavelength of interest, the acoustic intensity and power can be written as: Similarly, the acoustic energy density of the direct sound can be obtained as: Equations 5.28 and 5.29 provide the following relationship, that is, 21 (5.27) (5.28) (5.29) (5.30)

22. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field A similar relationship can be obtained for the reverberant sound. The reverberant sound energy can be regarded to be distributed in a closed space, which can be envisaged as the space surrounded by the surfaces of discontinuities that have various wall impedances. The total energy density comprising the direct and reverberant sound can therefore be written as 22 (5.31) (5.32) (5.33)

23. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.4 Direct and Reverberant Field The new parameter used in Equation 5.33 is This expresses the radius at which it is likely that the direct and reverberant sound participate equally. 23 (5.34, 35) Figure 5.6 Total energy density (the sum of direct and reverberant sound energy density; r 0, an intersection point, is the radius of reverberation)

24. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space Sound waves in a closed space can be regarded as the solutions that satisfy the boundary conditions of the closed space and the governing equation. There are two distinct approaches to acquire these solutions. The first is to obtain the solutions in the time domain, and the second is to acquire them in the frequency domain. In the frequency domain, it describes the sound waves in terms of the superposition of mode shapes. These approaches can be implemented by the following three methods. – The first regards the sound field of interest as the superposition of natural or normal modes that satisfy the boundary condition and the governing equation. – The second method describes the sound field using singular functions that satisfy the governing equation. – The latter method describes the sound field using acoustic rays, and is often referred to as ray acoustics. It assumes that the wavelength of interest is very much smaller than the characteristic length of the surface of reflection. 24

25. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space The latter method cannot be applied if the walls are no longer considered as locally reacting surfaces, or if the acoustic wavelength fails to meet the basic assumption of a locally reacting surface. (To get more information, see Section 3.9.1 from textbook.) 25 Figure 5.7 Conceptual example of ray acoustics

26. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space A sound field that falls into a given frequency within the closed space can be expressed by superposition of unique modes that meet the boundary condition and the governing equation, as where subscripts l, m, n refer to the respective orders of modes that correspond to individual coordinate directions of the Cartesian coordinate system. Let us consider a cube-shaped space in which sound can potentially be generated. Under the rigid wall boundary condition, where L x, L y, L z represent the lengths in each direction. 26 (5.36) (5.37)

27. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space In the case of relatively simple single dimension (i.e., a square tube with of length L ), A constant that represents the level of contribution that each unique mode makes to the entire sound field is called “modal coefficient”. To look at the behavior of modal coefficients in detail, let us observe sound fields that are radiated from a monopole sound source placed in a three-dimensional space. If the excitation is generated using a monopole sound source at the location of, 27 (5.38) (5.39) (5.40) (5.41) ( S : monopole amplitude)

28. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space Figure 5.8 depicts some individual modes contributing to the entire sound field, with each extent in a cubic room described by a given volume. 28 Figure 5.8 Sound pressure generated in a cubic space ( 0.8 ⅹ 0.6 ⅹ 0.1m 3 ); contributions by individual unique modes

29. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space Consider h lmn ( k ), a function that represents the frequency characteristics of a space. If the walls of a cubic room have the rigid body condition, k 2 lmn has a real-number value ( ) and is expressed as If the excitation frequency ( f = kc/2π ) of Equation 5.40 is the same as or similar to then the particular mode contribution ( a lmn ) will be infinite or significantly amplified. This frequency characteristic function, like the transfer function of a 1-DOF vibratory system, serves to adjust the extent of amplification for each mode depending on excitation frequency. 29 (5.42) (5.43)

30. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.5 Analysis Methods for a Closed Space The total number of participating modes and modal density increase dramatically as the frequency increases. In other words, a larger number of modes are needed to express sound fields as the frequency becomes higher. 30 Figure 5.9 Number of modes and modal density in a closed rectangular space ( L=5.8m; W=4.5m; H=2.5m ) with the walls being rigid bodies

31. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space An acoustically small space is one whose representative length or size is small relative to wavelength. An acoustically small space can generally be regarded as a vibratory system. A prime example of this is the Helmholtz resonator. 31 Figure 5.10 (a) Shape of simple resonator and equivalent vibratory system; (b) various types of resonator and conceptual samples; and (c) meaning of acoustic compliance

32. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space If the wavelength is considerably longer than the size of the resonator’s body and neck, the movements of fluid in the neck or the body will have almost identical phase. From Figure 5.10(a), the pressure change ( p in ) per unit time will reduce the volume change in the cavity of the resonator. If the pressure changes and volume are small enough to be linearized, Using acoustic compliance C A (which represents the volume change induced by unit sound pressure) as a proportional constant, When we have a large C A, the resonator undergoes a massive volume change. Equation 5.45 only highlights the correlation between pressure and volume. 32 (5.44) (5.45) (where p = p in ).

33. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space First of all, the volume change with respect to time in the cavity can be written as where u(t) is the velocity of fluid at the neck and A is the cross-sectional area of the neck. We can rewrite Equation 5.44 as Now consider the fluid motion at the neck. The balance between sound pressure acting on the fluid at the neck and the momentum of the fluid can be formulated as 33 (5.46) (5.47) (5.48) (5.49) ( l : length of the neck or effective length of the neck, to be more precise)

34. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space Substituting Equation 5.49 into Equation 5.47, we can obtain As noted before, p in = p ; Equation 5.50 can be rewritten as From Equation 5.51, the resonance radial frequency ( ω n ) can be obtained as 34 (5.50) (5.51) (5.52)

35. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space 35 (5.53) (5.54) (5.55) (5.56) (5.57) Neglecting higher order terms By the state equation, dp/dρ] s =c 2 (5.58) Equation 5.52 ( ) can be written as (5.59) ( m A : acoustic inertance)

36. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space The resonance frequency increases as the area of the neck becomes larger, but falls as the volume of the cavity becomes larger. This is because the wavelength associated with the resonance frequency is very long relative to the size of resonator. This causes the entire fluid at the neck to move in the same phase and the volume in the cavity to sustain the entire fluid at the neck as a kind of spring element. If a diameter of the neck is considerably smaller than the wavelength, the effective length (including end correction) of the neck can be expressed depending on whether it has a flange or not : To design the resonance frequency of a resonator precisely, the end correction factor should be taken into account. 36 (5.60) (5.61) l : length of the neck a : radius of cross-section

37. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space Neck and cavity are also basic components that consist of the geometrical shape of a resonator. In particular, impedance of a resonator can also be expressed as where Z r represents radiation impedance, and Z neck and Z cavity are impedances for the neck and the cavity, respectively. In particular, the reactance (imaginary part) of the impedance mainly determines resonance frequency: 37 (5.62) (5.63) can be obtained by open end correction can be derived under the assumption that the pressure in the cavity is maintained uniformly.

38. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space From observations in the book, which is omitted in this presentation, we can find that the geometry (the shape, location, and size of the neck and cavity) mainly affects the performance of a resonator. In addition, the shape of the neck is one of the main attributes which changes the absorption characteristics of resonator impedance. By changing the shape of the neck, we can therefore improve the absorption performance of a resonator. The necks can be any shape depending on practical requirements other than acoustical requirements. The shape is not very important if its spatial variation is considerably smaller than the wavelength of interest, such as the case of the neck of the Helmholtz resonator. 38

39. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space 39 Figure 5.12 A schematic model of a round resonator that has a gradually changing neck. L is the axial length of the cavity, l is the length of the neck, R is the radius of the cavity, r i is the inlet radius of the neck, and r o is the outlet radius of the neck We therefore consider a horn-shaped neck. The horn causes the impedance of propagating sound from a small source to gradually change to that of the impedance at the end of the horn, which lets the sound radiate well.

40. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space 40 Suppose that we have a plane wave propagating in the neck, then the wave is governed by Webster’s horn equation (see Section 5.7 for details). This can be written as where B = π(mx+r i ) 2, and m ( = (r o – r i )/l ) is the slope of the neck. r i, r o, and l are depicted in Figure 5.12. The solution is then where a 1 and a 2 are the magnitude of the incident and reflected wave, respectively. Particle velocity can be obtained by linerarized Euler’s equation, that is (5.77) (5.78) (5.79)

41. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space Then the impedance ( Z 0 ) at x=l can be written as Writing Equation 5.80 with respect to a 2 /a 1 yields In addition, the impedance ( Z i ) at x = 0 also can be obtained as 41 (5.80) (5.81) (5.82)

42. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space We can rewrite the impedance at the inlet of the neck ( Z i ) as If we assume that the wavelength of interest is much larger than the length of the neck, tan kl tends to kl. Equation 5.83 can then be simplified as We now examine the impedance at x = l ( Z o ). If fluid around the neck is moved about δ, the pressure change in the cavity can be expressed as 42 (5.83) (5.84) (5.85)

43. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space The impedance without regard to energy dissipation (resistance) at x = l can be written as Substituting Equation 5.86 into Equation 5.84, the impedance at the inlet of the neck ( Z i ) can be rewritten as where l denotes neck length that generally includes end correction. Therefore, the resonance frequency that sets reactance to zero can be obtained as 43 (5.86) (5.87) (5.88)

44. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.6 Characteristics of Sound in a Small Space A very common misunderstanding of a resonator is that it reduces sound by absorption. In reality, an abrupt impedance mismatch takes place at the resonance frequency of a resonator when installed on a noise transmission path (e.g. automotive engine suction/exhaustion units). This impedance mismatch reflects incident waves, and transmitted noise is finally reduced. In other words, it acts like an invisible wall. On the other hand, the amount of sound absorbed by a resonator is governed by its dissipation properties. The energy dissipation occurs primarily around the neck of the resonator, which is induced by friction between the fluid moving around the resonator’s neck and the confronting surface of the neck. The amount of dissipated energy, however, is generally much smaller than what is reflected by an impedance mismatch. 44

45. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics A duct is a space where the length of one direction is significantly greater than the cross-sectional direction. The sound propagation within a duct can be primarily expressed with respect to a single direction or coordinate. In the case of an infinite square duct as in Figure 5.13, 45 (5.89) (5.90) Figure 5.13 Three directions in which compressive fluids can move within an infinite duct

46. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics As noted on Figure 5.13, In terms of wavelength instead of wave number, Equation 5.93 can be rewritten as Equations 5.93 and 5.94 delineate the dispersion properties of a sound wave being propagated within a duct. 46 (5.91) (5.92) (5.93) (5.94)

47. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics k z, the propagation constant in the z direction, can be a real or imaginary number. – If it is a real number, it is propagated in the positive z direction. – If it is an imaginary number, the magnitude of sound waves attenuates exponentially as it progresses toward the propagation direction. (evanescent wave) In the wave number domain, only those modes whose wave numbers in the cross-sectional direction are lower than k=ω/c can propagate without being attenuated, that is The duct serves as a sort of low pass filter with the cut-off wave number of k. 47 (5.95)

48. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics If the cross-sectional area of a duct changes dramatically, this also significantly alters the way that a wave is propagated. 48 Figure 5.14 Propagation of waves in a wave guide with square section wave blocking wave tunneling

49. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics We now examine Equation 5.94 for a special case: the length in each sectional direction being shorter than half a wavelength. In this case, all modes in the sectional direction, excluding one where (m,n)=(0,0), will continue to be attenuated exponentially while being propagated. The only mode that is propagated without attenuation, (0,0), is a plane wave whose sound pressure remains constant in the sectional direction and whose wave number in the z direction is k. The wave in this case can be expressed as This implies that, if the characteristic length of a section is considerably smaller than the wavelength, the wave of a duct may be considered a one- dimensional problem. 49 (5.96)

50. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics Even in the absence of higher-order modes, massive changes takes place in the propagation of waves when the section experiences dramatic change. 50 Figure 5.15 Reflection and transmission of waves in simple divergent tube ( p i is an incident wave; p r a reflected wave; p st and p sr waves transmitted into and reflected by a silencer; and p t a transmitted wave)

51. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics These waves should meet the continuity condition at the planes whose sections are expanded ( z=0 ) and contracted ( z=L ), respectively. 51 (5.97) (5.98) (5.99) Figure 5.15(b) Reflection and transmission of waves in simple divergent tube ( p i is an incident wave; p r a reflected wave; p st and p sr waves transmitted into and reflected by a silencer; and p t a transmitted wave)

52. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics 52 At z=0, the pressure and the velocity need to be continuous, where S 1 and S 2 refer to the cross-sectional areas of the two tubes before and after expansion. The continuity condition at z=L can also be written as (5.100) (5.101) (5.102) (5.103)

53. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics On this basis, the magnitude ratio of transmitted waves against incident waves and transmission loss ( TL ), which indicates the power of incident waves being lost while passing through a silencer, are derived: The amounts of transmission and reflection are related to the sectional area and frequency of the two tubes. 53 (5.104) (5.105)

54. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 54 5.7 Duct Acoustics In Equation 5.105, transmission loss reaches its peak when sin kL has the highest value of 1; transmission losses becomes zero, which is the minimum value, when sin kL is zero. Figure 5.16 Comparison of transmission losses in simple divergent tube by wavelength (a) maximum transmission loss ( L=λ/4 ); and (b) minimum transmission loss ( L=λ/2 )

55. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics 55 Figure 5.17 Transmission loss by frequency and cross-sectional area ratio of simple divergent tube

56. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics A similar phenomenon occurs in a pipe of shape is illustrated in Figure 5.18. In this case, the length of the tube needs to be understood as the length of an effective tube, as described in Equation 5.60. Using an expansion chamber-based silencer, certain frequency elements in the noise of your choice can be reduced dramatically by adjusting the length of the expansion chamber. 56 Figure 5.18 Impedance mismatch due to duct resonance in 1/4 wavelength tube

57. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics A silencer that reduces noise using an impedance mismatch generated by a sharp change in shape is referred to as a reactive silencer or reactive muffler. One that tries to reduce noise using a perforated tube or sound-absorbing material is referred to as a dissipative silencer. In general, a dissipative silencer is known to be more effective for controlling high-frequency noise and absorbing noise better at relatively wider bandwidths. 57

58. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics As another special case, let’s look at an acoustic horn. Webster’s horn equation governs fluid particles in the acoustic horn. The acoustic horn can reduced reflection waves at the right end by slowly changing cross- section. 58 Figure 5.19 The relation between forces and the motion of an infinitesimal fluid element in an acoustic horn, which expresses momentum balance: forces (left-hand side) and momentum change (right-hand side). S denotes the cross-sectional area of the horn, u is the particle velocity in the x direction, ρ is volume density of the fluid, and ΔS is the ratio of the projected area of the area at x + Δx ( S x+Δx ) to the area at x ( S x )

59. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics The forces acting on the fluid between x and x+Δx and its motion will obey the conservation of momentum, that is where S represents the cross-sectional area of the horn, u is the particle velocity in the x direction, ρ is volume density of the fluid, and ΔS is the projected area of the area at x+Δx to the area at x. By neglecting higher-order terms and using the assumptions in Section 2.2, where p is the sound pressure, ρ 0 is the static volume density. 59 (5.106) (5.107)

60. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics The conservation of mass can be expressed as As Δx→0, By differentiating the right-hand side with respect to ρu and S, 60 (5.108) (5.109) (5.110) Figure 5.20 Conversation of mass in an infinitesimal fluid element in an acoustic horn. S denotes the cross-sectional area of the horn, u is the particle velocity in the x direction, and ρ is volume density of the fluid

61. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics With Equations 5.107 and 5.110, the state equation can, finally, provide us with Webster’s horn equation, that is, As a solution, where S 0 is the area at the left end of the horn, and α is a flare constant which expresses exponential increase as x becomes larger. 61 (5.107) (5.110) (5.111) (5.112)

62. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics Webster’s horn equation can then be written as The solution can be given by We can see the right-going waves are amplified as the waves propagate to the mouth. We can also obtain the phase velocity ( ) for the horn as which varies with frequency. 62 (5.113) (5.118) (Details can be found in the book.) (5.119)

63. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics There is a certain frequency which causes the phase velocity to be infinite, referred to as the cut-off frequency in the case of wave guides, that is 63 (5.120) Figure 5.21 Phase velocity ( c ) and cutoff frequency( f cutoff ) for an exponential acoustic horn, where c is the speed of sound in air ( 343 m/s at 20°C )

64. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics Suppose that we have a velocity source at the throat ( x = 0 ) with u 0 e -jωt ; we can easily obtain the pressure from the principle of momentum as By inserting the positive values of Equation 5.117 and Equation 5.120, we can obtain the radiation impedance of the exponential horn as When we have determined the cut-off frequency, the radiation impedance has a purely imaginary value ( -jρ 0 c ). This means that the waves in the horn cannot propagate well. 64 (5.121) (5.122)

65. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.7 Duct Acoustics 65 Figure 5.22 Driving point impedance of an exponential acoustic horn. The solid line denotes the real part of impedance (resistance), and the dashed line represents the imaginary part of the impedance (reactance) As frequency increases, the resistance term approaches the characteristic impedance of the medium ( ρ 0 c ) but the reactance term tends to 0. Note that the resistance term is 0 below the cut-off frequency.

66. Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 5.8 Chapter Summary The characteristics of sound generated in a relatively large space compared to the wavelength differ significantly from those of sound created in a smaller space. The sound generated in the former case can be considered to have direct and reverberant sound field. Reverberation time, suggested by Sabine, represents the properties of acoustically large space. In contrast to a larger space, the resonator properties of a small space are more similar to those of a 1-DOF vibratory system than its propagation properties, and this phenomenon can be utilized in controlling a variety of noises. If, as in the case of a duct, the characteristic length of its section is shorter than a wavelength and the length of its propagation direction is considerably longer than that of a wavelength, unique phenomena such as wave blocking and tunneling can be observed. 66