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

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Sound Propagation An Impedance Based Approach Acoustic Wave Equation and Its Basic Physical Measures Yang-Hann Kim Chapter 2

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Outline 2.1 Introduction/Study Objectives 2.2 One-dimensional Acoustic Wave Equation 2.3 Acoustic Intensity and Energy 2.4 The Units of Sound 2.5 Analysis Methods of Linear Acoustic Wave Equation 2.6 Solutions of the Wave Equation 2.7 Chapter Summary 2.8 Essentials of Wave Equations and Basic Physical Measures 2

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.1 Introduction/Study Objectives The governing equation is the total expression of every possible wave. This chapter explores the underlying physics and sensible physical measures that are related to acoustic waves. Impedance plays a central role with regard to its effect on these measures. The final objective of this chapter is to determine rational means of finding the solutions of acoustic wave equations. 3

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.2 One-dimensional Acoustic Wave Equation The simplest case is illustrated in Figure 2.1. 4 Figure 2.1 Relation between forces and motion of an infinitesimal fluid element in a pipe (expressing momentum balance: the left-hand side shows the forces and the right exhibits the change of momentum)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equations 2.2 and 2.3 are different simply because the displacement of the string has to be 0 at the end, but the acoustic pressure is maximal at. We can also predict that the driving point impedance is governed by (wave number multiplied by its length ). 2.2 One-dimensional Acoustic Wave Equation If the pipe is semi-infinitely long, then the pressure in the pipe can be mathematically written as (2.1) where is the pressure magnitude, and is an initial phase. 5 If the pipe is of finite length, then the waves in the pipe can be expressed by Equation 1.67. Recall that the displacement of the string, in this case, is (2.2) However, the possible acoustic pressure in the pipe can be written as (2.3)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd To understand what is happening in the pipe, we have to understand how pressures and velocities of the fluid particles behave and are associated with each other. As illustrated in Figure 2.1, the forces acting on the fluid between and and its motion will follow the conservation of momentum principle. That is, Sum of the forces acting on the fluid = momentum change. (2.4) We can mathematically express this equality as (2.5) where it has been assumed that the viscous force is small enough (relative to the force induced by pressure) to be neglected. 6 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The rate of change of the velocity can be expressed by (2.6) where is a function of position ( ) and time ( ) and the velocity is the time rate change of the displacement. Therefore, we can rewrite Equation 2.6 as (2.7) If the cross-section between and is maintained constant and becomes small, then Equation 2.5 can be expressed as (2.8) where (2.9), (2.10), (2.11) 7 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equation 2.11 is the total derivative, and is often called the material derivative. As can be anticipated, the second term is generally smaller than the first. If the static pressure ( ) and density ( ) do not vary significantly in space and time, then Equation 2.8 becomes (2.12) where is acoustic pressure and is directly related to acoustic wave propagation. Equations 2.8 and 2.12 describe three physical parameters, pressure, fluid density, and fluid particle velocity. In other words, they express the relations between these three basic variables. In order to completely characterize the relations, two more equations are needed. 8 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The relation between density and fluid particle velocity can be obtained by using the conservation of mass. Figure 2.2 shows how much fluid enters the cross-section at and how much exits through the surface at. If we apply the principle of conservation of mass to the fluid volume between and, the following equality can be written. the rate of mass increase in the infinitesimal element = the decrease of mass resulting from the fluid that is entering and exiting through the surface at and (2.13a) 9 2.2 One-dimensional Acoustic Wave Equation Figure 2.2 Conservation of mass in an infinitesimal element of fluid (increasing mass of the infinitesimal volume results from a net decrease of the mass through the surfaces of the volume).

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Expressing this equality mathematically leads to (2.13b) As assumed before, if the area of the cross-section ( ) remains constant, then Equation 2.13 can be rewritten as (2.14) We can linearize this equation by substituting Equation 2.10 into Equation 2.14. Equation 2.14 then becomes (2.15) Equations 2.12 and 2.15 express the relation between the sound pressure and fluid particle velocity, as well as the relation with the fluctuating density and fluid particle velocity, respectively. One more equation is therefore needed to completely describe the relations of the three acoustic variables. 10 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The other equation must describe how acoustic pressure is related to the fluctuating density. Recall that a pressure change will induce a change in density as well as other thermodynamic variables, such as entropy. This leads us to postulate that the acoustic pressure is a function of density and entropy, that is (2.16) where denotes entropy. We can then write the change of pressure, or fluctuating pressure, or, by modifying Equation 2.16 as follows : (2.17) 11 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equation 2.17 simply states that a pressure change causes a density change ( ) and entropy variation ( ). It is noticeable that the fluid obeys the law of isentropic processes when it oscillates within the range of the audible frequency. The second term on the right-hand side of Equation 2.17 is therefore negligible. Note that the second relation of Equation 2.18 is mostly found experimentally. This reduces Equation 2.17 to the form (2.18) where is the bulk modulus that expresses the pressure required for a unit volume change and is the speed of sound. 12 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd We can obtain Equation 2.18 by considering wave front propagation in a duct. Suppose that we make a disturbance which induces a small volume change in the one-dimensional duct as illustrated in Figure 2.21. We now want to find the relation between the speed of sound propagation and other physical variables, such as pressure and density. 13 2.2 One-dimensional Acoustic Wave Equation Figure 2.21 is cross-sectional area of the duct, is the coordinate that measures the distance from the disturbance, and is the disturbance velocity

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Conservation of mass implies the identity: (2.121) The left-hand side simply represents the amount of mass change per unit time due to the disturbance. The right-hand side is the mass flux of the fluid at rest. These two have to be balanced, and can be written as (2.122) We next apply Newton’s second law to the fluid of interest. The force difference will be, and the corresponding momentum change under consideration is which neglects higher order terms induced by. We can therefore write (2.123) Equations 2.122 and 2.123 lead to the following relation which describes the relation between the speed of propagation and other physical variables of fluid: (2.124) 14 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equation 2.124 states that the square of the speed of sound depends on the rate of compression with respect to density, that is, the amount of pressure requires to generate a unit change in density. Note, however, that the change in pressure and density of the fluid also depends on temperature or entropy. Therefore, Equation 2.124 has to be rewritten as (2.125) For isentropic process, Equation 2.125 can be written as (2.126) 15 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd For example, if the fluid can be assumed to be an ideal gas in isentropic process, then we can obtain the relations between pressure and density (the ideal gas law and the isentropic relation) as (2.127) (2.128) where is the number of moles defined as mass ( ) per unit molar mass ( ), is the universal gas constant (= 8.314 ) in standard air), is the absolute temperature ( ) and is the heat capacity ratio which is defined as the ratio of the specific heat capacity under constant pressure to the specific heat capacity under constant volume. Consequently, we can predict the speed of sound for an ideal gas as (2.129) under isentropic (i.e., no change in entropy) conditions. 16 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Tables 2.1 and 2.2 summarize the speed of sound in accordance with the state of gas. Table 2.1 Dependency of speed of sound on temperature 17 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Table 2.2 The dependency of the speed of sound on relative humidity and on frequency 18 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 19 2.2 One-dimensional Acoustic Wave Equation Figure 2.3 Pictorial relation between three variables that govern acoustic wave propagation ( and express the mean pressure and static density, respectively; and denote acoustic pressure and fluctuating density, respectively; denotes the speed of propagation, and is the velocity of the fluctuating medium)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd If we eliminate and from Equations 2.12, 2.15 and 2.18, then we obtain (2.19) Based on what we have studied so far, two conclusions can be drawn. – There is an analogy between the wave propagation along a string and acoustic waves, that is, the waves in compressible fluid. – There are definite relations between three acoustic variables, which are illustrated in Figure 2.3. 20 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd We now extend Equations 2.12, 2.15 and 2.19 to a three-dimensional case. First, Euler’s equation can be written as (2.20) where we use coordinate for convenience., and denote the velocity with respect to the coordinate system. We may use a vector notation to express Equation 2.20, which will yield a more compact form. This gives (2.21) where (2.22) 21 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equation 2.15 can also be extended to the three-dimensional form. That is, (2.23) The right-hand term of Equation 2.23 represent the net mass flow into the unit volume in space. If we eliminate and using Equations 2.21, 2.23 and 2.18, then (2.24) is obtained, which is a three-dimensional form of a wave equation. 22 2.2 One-dimensional Acoustic Wave Equation

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd For simplicity, we consider a one-dimensional case (Figure 2.4). We denote acoustic pressure ( ) as, and fluctuating density ( ) as. 23 2.3 Acoustic Intensity and Energy Figure 2.4 Volume change and energy for a one-dimensional element ( is potential energy density and is for convenience)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd As illustrated in Figure 2.4, there will be a volume change of because of the pressure difference along the element. The length of the element will be shortened by due to the small pressure change. The energy stored in the unit volume, potential or elastic energy, can then be written as (2.25) where has to obey the conservation of mass. We therefore have (2.26) Rearranging this, we obtain (2.27) 24 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Note that the last term on the right-hand side is much smaller than the others. Equation 2.27 therefore reduces to (2.28) Substituting Equation 2.28 into Equation 2.25 then gives (2.29) Using the state equation, Equation 2.18, and changing to, then gives (2.30) where denotes the acoustic potential energy. The kinetic energy per unit volume can be written as (2.31) 25 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd If we assume that the dissipated energy in the fluid is much less than the potential energy or kinetic energy, then the total energy has to be written:.(2.32) Note that the potential and kinetic energy are identical if the wave of interest is a plane wave in an infinite domain. The next question then is how the acoustic energy changes with time. We can see that the energy per unit volume has to be balanced by the net power flow through the surfaces that enclose the volume of interest, as illustrated in Figure 2.5. 26 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd This observation can be written conceptually as the rate of increase of energy = the power entering through the surface at ( ) (2.33) the power exiting through the surface at ( ). This can be translated into a mathematical expression as follows: (2.34) 27 2.3 Acoustic Intensity and Energy Figure 2.5 Relation between energy and one-dimensional intensity (energy in the volume and the intensity through the surface at and must be balanced).

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd Equation 2.34 can then be reduced to (2.35) where, which we call “acoustic intensity” or “sound intensity”. If we simply extend Equation 2.35 to a three-dimensional case, then (2.36) Two major points must be noted in relation to the expression of the intensity. – The intensity is a vector that has direction and magnitude. – The intensity is a product of two different physical quantities. 28 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd When we have two physical variables, the phase difference between them has significant meaning. The phase between the force and velocity expresses how well the force generates the velocity (response). In this regard, the intensity can be classified as two different categories: active intensity and reactive intensity. To understand the meaning of the intensities in physical terms, we look again at the simplest case: the intensity of waves propagating in a one- dimensional duct. Figure 2.6 depicts the waves in an infinite-length duct and Figure 2.7 shows the waves for a finite-length duct. 29 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 30 2.3 Acoustic Intensity and Energy Figure 2.6 The acoustic pressure and intensity in an infinite duct. Note that the pressure and velocity are in phase with each other. Also, the active intensity (or average intensity with respect to time) is constant

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 31 2.3 Acoustic Intensity and Energy Figure 2.7 The acoustic pressure and intensity in a duct of finite length of. Note that the phase difference between the pressure and velocity is 90°( ))

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd When the waves propagate in an infinite duct, the pressure and velocity have the same phase. – It can be observed that the average intensity with respect to time is constant, as can be seen in Figure 2.6. – The instantaneous intensity changes with regard to the position along the duct. – The excitation effectively supplies energy to the system. If we have the same excitation at one end, the duct has a finite length of, and a rigid boundary condition exists at the other end, then the phase difference between the pressure and velocity will be 90〫( ). – It is not possible to effectively put energy into the system. – The intensity is always zero at the nodal point of the duct, but it oscillates between these points where the energy vibrates and does not propagate anywhere. 32 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd We now need to explore more specific characteristics of the sound intensity, such as how to calculate and measure the intensity. The mathematical definition of intensity can be written as (2.37) The one-dimensional expression is simply (2.38) The velocity can be obtained from the Euler equation (2.12): (2.39) To obtain the derivative with respect to space, we may use two microphones. This means that we approximate the derivative as (2.40) 33 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The pressure ( ) at the position of the measurement can be approximately obtained as: (2.41) where the pressure fluctuates in time and is therefore a dynamic quantity. We now look at the intensity measurement and calculation by considering a plane wave with a radian frequency. The pressure can then be written as (2.42) where denotes the pressure magnitude which has a real value and represents the possible phase change in space. 34 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd To obtain the velocity using the linearized Euler equation a pressure gradient is needed, that is (2.43) Equations 2.39 and 2.43 then allow us to obtain the particle velocity: (2.44) The intensity that is generated by the real part of pressure (2.42) and the corresponding velocity (2.44), which is in phase with the real part of the pressure, can be obtained as (2.45) This is normally referred to as the “active component of sound intensity”. 35 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The time average of this intensity is often referred to as a mean intensity, or an active intensity, and can be written: (2.46) This intensity can effectively supply power to space. On the other hand, the multiplication of the real part of the pressure and the imaginary part of the velocity that has 90〫phase difference will generate the following intensity: (2.47) We refer to this intensity as the “reactive component of sound intensity”. The time average of this intensity is 0 and, therefore, there is no net energy transport; it only oscillates. 36 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The directions of intensities – The direction of the active intensity is perpendicular to the wave front where the phase is constant. – The direction of reactive intensity has to be perpendicular to the surface over which the mean square pressure is constant. Equations 2.45 and 2.47 are referred to as the instantaneous active intensity and instantaneous reactive intensity in the strict sense, respectively. However, – when we say active intensity, we are referring to a time average of the instantaneous active intensity, that is, (2.46) – for the reactive intensity case, we call its amplitude (2.48) as reactive intensity. 37 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd The instantaneous intensity expressed by Equation 2.38 is composed of two components : the instantaneous active intensity (2.45) and instantaneous reactive intensity (2.47). We can therefore write them as (2.49) Using a complex function, Equation 2.49 can be expressed in simpler form, that is (2.50) where. This is often referred as a complex intensity. 38 2.3 Acoustic Intensity and Energy

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound The units that are relevant to sound can be classified into two groups: absolute units and subjective units. Absolute units – The unit of pressure = Pascal(Pa) = N/m 2 – The unit of velocity = m/sec – The unit of intensity = Pa‧m/sec = watt/m 2 – The unit of energy = joule = watt‧sec To understand subjective units, we need to understand how we hear. This means that we need to study our hearing system. Figure 2.8(a) depicts the human hearing system. 39

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound 40 Figure 2.8 The structure of the ear and its frequency band characteristics. (a) The structure of a human ear. (Redrawn with permission from D. Purves et al., Neuroscience, 3rd edition, 2004, pp. 288 (Figure 12.3), Sinauer Associates, Inc., Massachusetts, USA. Ⓒ2004 Sinauer Associates, Inc.) (b) External, middle, and inner ear. (c) Basilar membrane and Corti organ. (d) The cross-section of the cochlea shows the sensory cells (located in the organ of Corti) surrounded by the cochlear fluids. (e) Space-frequency map: moving along the cochlea, different locations are preferentially excited by different input acoustic frequencies. (f) Tonotopic organization. (Figure 2.8(b–f) drawings by Stephan Blatrix, from “Promenade around the cochlea,” EDU website: http://www.cochlea.org by Re my Pujol et al., INSERM and University Montpellier.)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound It is well known that humans do not hear the frequency of sound in absolute scale, but rather relatively. Due to this reason, we normally use relative units for the frequencies. The octave band is a typical relative scale (Figure 2.9). 43 Figure 2.9 Octave, 1/3 octave, and 1/n octave The center frequency ( ) of each band is at the geometrical center of the band.

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound For the amplitude of the sound pressure, we use the sound pressure level (SPL or ). It is defined as (2.51) and is measured in units of decibels (dB); is the reference pressure, is the average pressure, and log 10 is a log function that has a base of 10. is 20 Pa(20×10 －6 N/m 2 ). From Figure 2.10, we can see that the human can hear from about 0 dB to somewhere in the range of 130-140 dB. Table 2.4 collects some typical samples of sound levels that we can encounter, providing some practical references of the sound pressure level (SPL). 45

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound 46 Figure 2.10 Equal-loudness contour: each line shows the SPL with respect to the frequency which corresponds to a loudness (phon) of 1 kHz pure sound. (Reproduced from ISO 226 (2003), “Normal equal-loudness-level contours,” International Standards Organization, Geneva.)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound In order to calculate SPL, we write (2.52) where denotes the measurement time. Equation 2.52 expresses as the sum of every frequency component, equivalent to (2.53) We then use the well-known relation (2.54) where * denotes the complex conjugate. 48

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound (2.55). This is only valid if and only if Equation 2.55 has a maximum when. – When, the slowly fluctuating terms with frequency are much greater than those with a frequency of. 49 If we rearrange Equation 2.53 using Equation 2.54, then we obtain

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound 50 Figure 2.11 Total mean square pressure and the mean square pressure of each frequency band Figure 2.11 illustrates the relation between the sound pressure level and the mean square pressure.

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound Let us begin with two sound pressures that have different frequencies, and. According to Equation 2.51, the sound pressure level of each individual tone can then be written as (2.56) (2.57) If these two tones occur at the same time, the SPL can be written (2.58) If we generalize this result to different pure tone cases, the SPL is (2.59) 51

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd For example, if each tone has an SPL of 80dB, that is, =80dB and =80dB, then the sum of these two must be. This simply means that the SPL increases by 3dB. If we have two sounds of SPL 75dB and 80dB, the resulting SPL of the sounds is 81.2dB. 52 2.4 The Units of Sound

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.4 The Units of Sound As illustrated in Figure 2.10, our hearing system depends strongly on frequency band. Therefore, SPL has to properly consider its effect. Figure 2.12 shows typical weightings or scales : often used as dB(A,B,C) 53 Figure 2.12 Various weighting curves. A-weighting: 40 phon curve (SPL 85 dB).

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation This section addresses how we mathematically predict or describe sound in space and time. Let us begin with the case of which we have a volume source in one- dimensional infinite space. The volume velocity source makes the mass change by the velocity excitation. That is (2.60) where is the volume velocity at. 55

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation Substituting Equations 2.18 and 2.12 into this new mass law Equation 2.60, we obtain the governing equation that includes the acoustic source: (2.61) We first attempt a harmonic solution, (2.62) Equation 2.61 can then be written as (2.63) whererepresents the right-hand side of Equation 2.61. Equation 2.63 is strictly only valid where the sound source exists; otherwise a homogeneous equation is valid. 56

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation For example, if there is a point source at, then Equation 2.63 can be rewritten as (2.64) where is a Dirac delta function, that is (2.65) If the source exists only in the region, then we can write the governing equation as (2.66) 57

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation Expanding this equation to a three-dimensional case yields (2.67) where and express the source position and the volume where the source is, respectively. We now look at how to mathematically express the boundary condition. We first study the one-dimensional case. As already expressed (1.26), the boundary condition can generally be written as (2.68) where the subscript 0 and represent boundary at. 58

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation To understand the boundary conditions that are expressed by Equation 2.68, let us investigate several typical cases. First, when, the condition takes the form (2.69) This type of boundary condition is generally known as the Dirichlet boundary condition. On the other hand, if, then the equation becomes (2.70) This type of boundary condition is called the Neumann boundary condition. 59

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation If, then Equation 2.68 reduces to (2.71) and (2.72) The impedance is described on the boundary. More generally, the three-dimensional case of Equation 2.71 can be written as (2.73) where is the surface that encloses the space of interest, as depicted in Figure 2.13 where is the particle velocity that is normal to the surface. 60

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation We will consider the problem that is governed by the inhomogeneous governing equation and homogeneous boundary condition. That is, (2.66) and (2.71) 61 Figure 2.13 General boundary value problem ( is complex amplitude, is complex velocity, is the wave number, and S expresses the boundary; and indicate the observation position and boundary, respectively)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation One very well-known method for obtaining the solutions which satisfy Equations 2.66 and 2.71 uses eigenfunctions to express the solution. This means, especially, that we first try to find the function which satisfies (2.74) and also satisfies the boundary condition of Equation 2.71; that is (2.75) The function which satisfies Equations 2.74 and 2.75 is the eigenfunction or eigenmode, and the constant is the eigenvalue. To shed more light on this problem, we consider the special case when. In this case, we have a rigid-wall boundary condition and the eigenmode can be found, intuitively, as: (2.76) 62

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation If, which is the case for the pressure release boundary condition, then the solution has to take the form (2.77) We generally call this method, which attempts to obtain the solution by superimposing the eigenfunctions, a modal analysis. The advantage of this method is that a linear combination of the eigenmodes also satisfies the given boundary condition. For the one-dimensional case, the pressure can be written as (2.78) 63

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation The main obstacle in finding the solution by a linear combination of eigenfunctions is finding each mode’s contribution, or weighting, on the solution. In other words, we must attempt to find (2.78) that satisfies Equation 2.66. For example, if we have one source at a point where Equation 2.64 is the governing equation, then we can attempt to construct the solution as given by Equation 2.78. The coefficients can be found by using the property of the eigenfunctions (orthogonality condition), that is (2.79) and (2.80) 64

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation Using Equations 2.66, 2.74, 2.78, 2.79 and 2.80, we can obtain the weighting as (2.81) where denotes the complex conjugate. Alternatively, we can try to obtain the solution that satisfies the boundary condition by introducing Green’s function. If we denote the sound pressure due to a unit point source at as, then has to satisfy the equation: (2.82) 65

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation Multiplying by Equation 2.63 and by Equation 2.82, subtracting the former from the latter and finally integrating with respect to lead us to (2.83) Then, integration by parts yields: (2.84) Changing the variable to reduces Equation 2.84 to the form (2.85) We now investigate how to apply Equation 2.85 when we have a unit amplitude sound source at, as illustrated in Figure 2.14. This specific case reduces Equation 2.85 to (2.86) 66

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation 67 Figure 2.14 One-dimensional and three-dimensional boundary value problems

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation If the velocity at is 0 (rigid-wall boundary condition), or the pressure is 0(pressure release boundary condition), then Equation 2.86 becomes (2.87) or (2.88) Equation 2.86 states that the sound pressure at consists of two components: one is a direct effect from the sound source and the other is due to the reflection from the boundary. Expanding Equation 2.86 to a three-dimensional form yields the integral equation (2.89) 68

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.5 Analysis methods of linear Acoustic Wave Equation If we do not have the sound source in the integral volume (Figure 2.14(b)), then Equation 2.89 becomes (2.90) Equations 2.89 and 2.90 are referred to as Kirchhoff-Helmholtz integral equations. 69

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation We will start with a one-dimensional, planar acoustic wave at position and time,. This can be written as (2.91) A wave in a certain direction in space can be expressed as (2.92) where is a complex amplitude. The plane wave 2.92, as the name implies, has all the same physical properties at the plane perpendicular to at (Figure 2.15). Note that its impedance at any position and time is (2.93) 70

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation The plane wave in an unbounded fluid propagates in the wave number vector direction, independent of the position, frequency, wave number, and wavelength. Intensity, specifically the average intensity (active intensity), can be expressed as follows : (2.94) where is the velocity in the direction of propagation and 71 Figure 2.15 A plane wave ( is normal to the planes of constant phase)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation Therefore, the intensity (2.94) can be written as (2.95) The governing equation can also be written in terms of the spherical coordinate. We assume that the pressure is independent of the polar and azimuth angles and only depends on the distance from the origin ( ). Equation 2.24 then becomes (2.96) Its solution will be (2.97) where is a complex amplitude. 72

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation Equation 2.97 can be rewritten as (2.98) To assess the velocity, consider Euler’s equation in the spherical coordinate: (2.99) where is the velocity in the radial direction. Equations 2.98 and 2.99 allow us to calculate the velocity in the radial direction, that is (2.100) Therefore, the impedance at can be written as (2.101) This is the monopole radiation impedance. 73

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation 74 Figure 2.16 depicts Equation 2.101. Figure 2.16 Monopole radiation. (a) The monopole’s radiation impedance where is wave number, indicates the Note is noteworthy that it behaves as a plane wave, as the observation position is far from the origin. (b) Pressure and particle velocity in near field ( is small), magnitude (left) and phase (right) of pressure (top) and particle velocity (bottom); arrows indicate intensity. (c) As for (b) for far field case

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation In the near field, the reactive part dominates the acoustic behavior in such a way that the waves do not propagate well in the vicinity of the origin. In the far field, the active part dominates. Therefore, the wave propagates as if it is a planer wave. The monopole sound source is defined by Equations 2.98 and 2.100 and has a singularity at. This simple solution satisfies the linear wave equation. This implies that superposition of this type of solution also satisfies the governing wave equation. We can therefore attempt to construct any type of wave by using the monopole. This concept is illustrated in Figure 2.17, a graphical expression of Huygen’s principle. 75

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation 76 Figure 2.17 Huygen’s principle. The wave front constructed by many monopole sound sources: (a) graphical illustration and (b) shallow ripple tank

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.6 Solutions of the Wave Equation If the two monopoles are close together with the opposite phase, then a dipole is formed. 77 Figure 2.18 Dipole and quadrupole distributions and their characteristics where indicates an arbitrary point in spherical coordinate, is wave number, represents the dipole-moment amplitude vector, and represents the amplitude of quadrupole: (a) pressure of the spatial pattern of dipole sound; (b) impedence of a dipole at ; (c) magnitude (left) and phase (right) of particle velocity of a dipole in near field (top) and far field (bottom) (arrows indicate intensity); (d) pressure of a quadrupole pattern in space; (e) impedance of a quadrupole at and ( ) magnitude (left) and phase (right) of particle velocity of a quadrupole in near field (top) and far field (bottom) (Section 2.8.4)

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 2.7 Chapter Summary We have attempted to understand how acoustic waves are generated and propagated in a compressible fluid. Conservation of mass and the state equation of fluid, together with Newton’s law, provide three relations between density, fluid velocity, and pressure. Acoustic intensity expresses the direction of acoustic power flow as well as its magnitude. We studied a way to measure the associated acoustic variables in accordance with human perception. We have investigated possible solution methods that predict how sound waves propagate in space and time. 80

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

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

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