Chapter 2 Acoustical Background

Transcription

1 Chapter 2 Acoustical Background Abstract The mathematical background for functions defined on the unit sphere was presented in Chap. 1. Spherical harmonics played an important role in presenting and manipulating these functions. In this chapter, functions on the sphere are defined through the formulations of fields in three dimensions. Although sound fields are of primary concern in this book, which is oriented towards microphone arrays, the material presented here can be applied to scalar fields in general. This chapter begins by presenting the acoustic wave equation in Cartesian and spherical coordinates, with possible solutions. Solutions to the wave equation in spherical coordinates are shown to involve spherical harmonics and spherical Bessel and Hankel functions. Having formulated the fundamental solutions, sound fields due to a plane wave and a point source are presented, including an analysis of the effect of a rigid sphere introduced into the sound field. The latter is useful for describing the sound field around a microphone array configured over a rigid sphere, for example. The chapter concludes with a formulation of the three-dimensional translation of sound fields. 2.1 The Acoustic Wave Equation Sound pressure in a free three-dimensional space, denoted by p(x, t), and measured in Pascals, with x = (x, y, z) R 3 measured in meters, and t representing time in seconds, satisfies the homogeneous acoustic wave equation [25]: x 2 p(x, t) 1 2 p(x, t) =, (2.1) c 2 t2 with c denoting the speed of sound in air, typically 343 m/s under normal ambient conditions, and x 2 denoting the Laplacian in Cartesian coordinates, defined for a function f (x, y, z) as x 2 f 2 x f y f z f. (2.2) 2 The erratum of this chapter can be found under DOI 1.17/ _8 Springer-Verlag Berlin Heidelberg 215 B. Rafaely, Fundamentals of Spherical Array Processing, Springer Topics in Signal Processing 8, DOI 1.17/ _2 31

2 32 2 Acoustical Background For a single-frequency sound field, the sound pressure can be expressed as p(x, t) = p(x)e iωt, (2.3) where ω is the radial frequency in radians per second. In this representation, p(x) can be regarded as the space-dependent amplitude of the sound pressure at frequency ω. With k = ω/c denoting the wave number in radians per meter, the dependence of the pressure amplitude on ω or on k can be explicitly described using the notation p(k, x). Substituting Eq. (2.3) into Eq. (2.1), the wave equation transforms into the Helmholtz equation (with time-dependence omitted): 2 x p(k, x) + k2 p(k, x) =. (2.4) The notation p(k, x) can also be used to represent broadband or multiple-frequency sound fields in steady state, in which case p(k, x) is the Fourier transform of the sound pressure at frequency ω = k/c. Note that p is complex, representing the complex amplitude of the sound pressure. The actual sound pressure, such as would be measured by a microphone, for example, is given by the real part of p. A solution to the wave equation can be derived by using separation of variables. The most commonly used solution is the plane wave, given by p(x, t) = Ae ik x e iωt, (2.5) where A is the amplitude, k (k x, k y, k z ) represents the wave vector, denoting the propagation direction of the plane wave, and k x = k x x+k y y+k z z represents the dot product of vectors k and x. Plane wave sound fields can be described directly using this solution. Representation of other sound fields is also possible using the Fourier transform, with e ik x providing the basis function for describing the spatial variation of the sound pressure amplitude. In some cases it may be more useful to denote the direction of arrival of the plane wave, rather than the direction of propagation. For this purpose, wave vector k = k is introduced, denoting the arrival direction, and will be used later in this chapter. In this case the sound pressure is given by p(x, t) = Ae i k x e iωt. (2.6) In this book, sound fields are measured by spherical microphone arrays, so that it is preferable to represent the position vector in spherical coordinates, r = (r,θ,φ). The wave equation is now rewritten in spherical coordinates, for which the Laplacian in spherical coordinates is first defined for a function f (r,θ,φ): r 2 f 1 (r 2 r ) r 2 r f + 1 ( sin θ ) r 2 sin θ θ θ f + 1 r 2 sin 2 θ 2 φ 2 f. (2.7) Equation (2.7) can be derived from the Laplacian in Cartesian coordinates [Eq. (2.2)] using Eq. (1.4) and the chain rule. The wave equation in spherical coordinates can

3 2.1 The Acoustic Wave Equation 33 now be written as 2 r p(r, t) 1 c 2 2 p(r, t) =, (2.8) t2 where p(r, t) is the sound pressure as a function of time and space in spherical coordinates. For single frequency, i.e. harmonic, sound fields, the Helmholtz equation can also be written in spherical coordinates as 2 r p(k, r) + k2 p(k, r) =, (2.9) where p(k, r) is the amplitude of the sound pressure over space and the dependence on k is explicitly expressed. The amplitude of the pressure can be represented in a way similar to Eq. (2.3), as p(r, t) = p(r)e iωt. (2.1) A solution to the wave equation (2.8) can be obtained using separation of variables: p(r, t) = R(r)Θ(θ)Φ(φ)T(t). (2.11) Substituting Eq. (2.11) into the wave equation (2.8), the single equation as a function of p can be decomposed into four partial equations in the separate variables Θ,Φ,R and T. The equation representing dependence on time is a second-order differential equation: d 2 T dt + 2 ω2 T =, (2.12) with a fundamental solution T(t) = e iωt, ω R, (2.13) as also implied by Eq. (2.1). Substituting Eq. (2.11) in the Helmholtz equation (2.9) and multiplying by r 2 sin 2 θ/p, the term depending on φ can be isolated, satisfying with a fundamental exponential solution d 2 Φ dφ 2 + m2 Φ =, (2.14) Φ(φ) = e imφ, m Z, (2.15) where m is an integer because of the periodicity of Φ as a function defined over the unit circle, with φ [, 2π). Substituting Eq. (2.15) back into the Helmholtz equation, a term dependent only on θ can be isolated, satisfying d dμ [ (1 μ 2 ) d dμ Θ ] + [n(n + 1) m2 1 μ 2 ] Θ =, (2.16)

4 34 2 Acoustical Background with μ = cos θ. This equation is known as the associated Legendre differential equation and has two types of solutions, one singular at μ = 1 and a second solution that is typically selected and is referred to as the associated Legendre function of the first kind: Θ(θ) = Pn m (cos θ), n N, m Z. (2.17) Substituting Eq. (2.16) into the Helmholtz equation and applying some further manipulations, a term dependent only on r can be isolated, satisfying ρ 2 d2 dρ 2 V + 2ρ d dρ V + [ ρ 2 n(n + 1) ] V =, (2.18) with ρ kr and V(ρ) R(r). This equation is known as the spherical Bessel equation and its solution comprises spherical Bessel functions of the first kind, j n (kr), or spherical Hankel functions of the first kind, h n (kr), or both (see Sect. 2.2). Combining the solutions over r, θ, φ, and t, a fundamental solution for the wave equation in spherical coordinates can written in the form p(r, t) = j n (kr)y m n (θ, φ)eiωt (2.19) or p(r, t) = h n (kr)y m n (θ, φ)eiωt, (2.2) or as a combination of these solutions for various values of n and m. Specific solutions in the case of a plane-wave sound field and a sound field produced by a point source are presented later in this chapter. 2.2 Spherical Bessel and Hankel Functions Solutions to the wave equation in spherical coordinates include spherical Bessel and Hankel functions. These functions are presented in this section. The spherical Bessel function of the first kind, j n (x), and of the second kind, y n (x), can be written using Rayleigh formulas as [4] j n (x) = ( 1) n x n ( 1 x ) d n sin(x) dx x (2.21) and y n (x) = ( 1) n x n ( 1 x ) d n cos(x), (2.22) dx x with the spherical Hankel functions of the first kind, h n (x), and the second kind, h n (2)(x), written as ( ) 1 h n (x) = i( 1) n x n d n e ix (2.23) x dx x

5 2.2 Spherical Bessel and Hankel Functions 35 and with the relations and ( ) 1 h n (2) (x) = i( 1)n x n d n e ix x dx x, (2.24) h n (x) = j n (x) + iy n (x) (2.25) h (2) n (x) = j n(x) iy n (x). (2.26) Because the spherical Bessel functions are real, j n (x) and y n (x) compose the real and imaginary part of h n (x), i.e. j n (x) = Re {h n (x)} (2.27) and y n (x) = Im {h n (x)}. (2.28) The spherical Bessel and Hankel functions are also related to the Bessel function, J α (x), and the Hankel function, H α (x), through π j n (x) = 2x J n+ 1 (x) (2.29) 2 and π h n (x) = 2x H n+ 1 (x). (2.3) 2 Solutions to the wave equation can be represented as a linear combination of spherical Bessel functions of the first and second kinds, or of spherical Bessel and Hankel functions. The latter representation is more common and will be employed in this book. Tables 2.1 and 2.2 present expressions for the spherical Bessel function and the spherical Hankel function of the first kind, respectively, for the first few orders. Figures 2.1 and 2.2 illustrate j n (x) for the first few orders. Table 2.1 Spherical Bessel functions of the first kind for n =,...,3 j (x) = sin x x j 1 (x) = cos x x + sin x x 2 j 2 (x) = sin x x j 3 (x) = cos x x 3cosx x 2 6sinx x 2 + 3sinx x 3 15 cos x x sin x x 4

6 36 2 Acoustical Background Table 2.2 Spherical Hankel functions of the first kind for n =,...,3 h (x) = eix ix h 1 (x) = eix (i+x) x 2 h 2 (x) = ieix ( 3+3ix+x 2 ) x 3 h 3 (x) = eix ( 15i 15x+6ix 2 +x 3 ) x 4 Figure 2.2 shows that at low values of x, j n (x) has a steeper slope for larger orders. Indeed, j n (x) for x 1 can be approximated by [4] Fig. 2.1 Magnitude of the spherical Bessel function, j n (x),forn =,..., Fig. 2.2 Magnitude of the spherical Bessel function, j n (x),forn =,...,6andforx < 1

7 2.2 Spherical Bessel and Hankel Functions Fig. 2.3 Magnitude of the spherical Hankel function, h n (x),forn =,...,6 j n (x) x n, (2n + 1)!! x 1, (2.31) where ( )!! is the double factorial function, e.g. (2n + 1)!! = (2n + 1)(2n 1) 1. Figure 2.1 shows that at large values of x, the amplitude of j n (x) decays in a similar manner for all n. Indeed, for x n [or more specifically, for x n(n + 1)/2] j n (x), as expressed in Table 2.1, is dominated by the first term, decays as 1/x, and can be approximated by [4] j n (x) 1 sin(x nπ/2), x n(n + 1)/2. (2.32) x Figure 2.1 also shows that the spherical Bessel function has zeros. The zeros of j (x) are at ±lπ, l = 1, 2,... ; for higher orders, the first zeros are positioned at x >π, but tend to appear at a spacing of π for large x, as suggested by Eq. (2.32). Figure 2.3 presents h n (x), illustrating that the spherical Hankel functions, unlike the spherical Bessel functions, diverge towards the origin. Furthermore, Fig. 2.4 illustrates that for x 1, higher orders increase towards the origin with a larger slope. This is supported by the small argument approximation of the spherical Hankel function: (2n 1)!! h n (x) i, x 1. (2.33) x n+1 On the other hand, for large values of x, h n (x) decays similarly for all n, which is supported by the large argument approximation: h n (x) ( i) n+1 eix, x n(n + 1)/2. (2.34) x

8 38 2 Acoustical Background Fig. 2.4 Magnitude of the spherical Hankel function, h n (x),forn =,...,6andx < 1 The spherical Bessel function also satisfies recurrence equations: 2n + 1 j n (x) = j n 1 (x) + j n+1 (x) (2.35) x and (2n + 1)j n (x) = nj n 1(x) (n + 1)j n+1 (x), (2.36) with j n (x) denoting the first derivative of j n(x) with respect to x. These relations also hold for the spherical Bessel function of the second kind and the spherical Hankel functions of the first and second kinds [4]. 2.3 A Single Plane Wave Consider a unit-amplitude, single-frequency plane wave, arriving from direction (θ k,φ k ) with a wave vector k = k = (k,θ k,φ k ) written in spherical coordinates. The plane wave is a solution to the homogeneous wave equation in Cartesian coordinates, and so could also be represented as a combination of the general solutions of the wave equation in spherical coordinates. As the spherical Hankel functions diverge at the origin, spherical Bessel functions are used to represent a plane-wave sound field. The familiar expression for the sound pressure at r = (r,θ,φ) due to a plane wave, i.e. e ik r, can be written as a summation of spherical harmonics and spherical Bessel functions [23, 56]: p(k, r,θ,φ) = e ik r = e i k r = 4πi n j n (kr) [ Yn m (θ k,φ k ) ] Y m n (θ, φ). (2.37) n= m= n

9 2.3 A Single Plane Wave 39 The dot product is given by k r = kr cos Θ. By applying the spherical harmonics addition theorem, as in Eq. (1.26), Eq. (2.37) is reduced to p(k, r,θ)= e ikr cos Θ = i n j n (kr)(2n + 1)P n (cos Θ). (2.38) n= The exponential representation of a plane wave, as in the first line of Eq. (2.37), is simple and natural, compared to the infinite summation on the second line of the same equation. However, the advantage of representing a plane wave in spherical harmonics lies in the possibility of performing separation of variables. Terms including kr, wave arrival direction (θ k,φ k ), and position (θ, φ) on the surface of a sphere of radius r, can thus be formulated as parameters of separate functions. This advantage is exploited later in the book when developing array processing algorithms in the spherical harmonics domain. Derivation of Eqs. (2.37) and (2.38) and further reading can be found in [4, 56], for example. The shortcoming of the representation of plane waves using spherical harmonics with an infinite summation is typically overcome in practice by approximating the infinite summation with a finite summation, i.e. Eq. (2.37) is rewritten as p(k, r, θ, φ) N n= m= n 4πi n j n (kr) [ Yn m (θ k,φ k ) ] Y m n (θ, φ), (2.39) introducing truncation errors. As an example of spherical harmonics representation of a plane-wave sound field, consider a sound field composed of a single unit-amplitude plane wave arriving from (θ k,φ k ) = (9, 2 ). Figure 2.5 shows the real part of the sound pressure, Re{p},for k = 1, computed using Eq. (2.39) for various values of N and plotted over the xy plane. The figure shows that for N = 32, a sinusoidal behavior is observed, as expected from the real part of the amplitude of a single plane wave. However, for smaller values of N, e.g. N = 16 and N = 8, the sinusoidal behavior is distorted, and is only maintained within a limited circle around the origin. This behavior is typical of the representation of plane waves using spherical harmonics it is accurate only within the volume of a sphere. The radius of the sphere depends on k and N, as discussed next. Equation (2.37) provided an expression for the sound pressure at (r,θ,φ) for a sound field composed of a single plane wave. Now, the sound pressure is evaluated at the surface of a sphere of radius r. Therefore, p(k, r,θ,φ)is a function defined over a sphere, having a spherical Fourier transform with coefficients denoted by p nm (k, r) satisfying p(k, r, θ, φ) = p nm (k, r)yn m (θ, φ). (2.4) n= m= n

10 4 2 Acoustical Background Fig. 2.5 Re{p(k, r,θ,φ)} for a unit-amplitude plane wave arriving from (θ k,φ k ) = (9, 2 ) and computed using Eq. (2.39) with N = 8, 16, 32 and k = 1, plotted over the xy plane

11 2.3 A Single Plane Wave Fig. 2.6 Magnitude of the normalized spherical Bessel function, 4πi n j n (kr),forkr = 8, 16 Comparing Eqs. (2.37) and (2.4), the spherical harmonic coefficients for the sound pressure over a sphere of radius r, in a sound field composed of a single unit-amplitude plane wave arriving from (θ k,φ k ), can be written as p nm (k, r) = 4πi n j n (kr) [ Y m n (θ k,φ k ) ]. (2.41) Equation (2.41) also shows that the magnitude of p nm is proportional to the magnitude of j n (kr). It is therefore expected that p nm for a plane-wave sound field decays as a function of n for n > kr, as suggested by Fig. 2.1, and, more explicitly, as illustrated in Fig. 2.6 for kr = 8 and kr = 16. This is an important result, as it suggests that the sound field represented by the infinite summation in Eq. (2.37) can be represented by

12 42 2 Acoustical Background a finite summation as in Eq. (2.39) with little error. The spherical harmonics series for a plane-wave sound field can therefore be considered as nearly order limited, so that sampling theories for order-limited functions, as detailed in Chap. 3, can be applied with little error. This behavior is clearly illustrated in Fig. 2.5 for N = 16, for example. The figure shows a circle of radius r = 16 (equivalent to kr = 16 because k = 1), illustrating that with N = 16, the pressure inside the circle satisfying kr < N is reconstructed accurately, while outside the circle, with kr > N, the pressure is reconstructed with significant error. The condition of kr < N for accurate sound pressure reconstruction is further illustrated in the following example, analyzing the magnitude of sound pressure over the surface of a sphere of a fixed radius r, at wave number k, satisfying kr = 1. The pressure is produced by a single unit-amplitude plane wave arriving from direction (θ k,φ k ) = (45, 45 ), which is then reconstructed using Eq. (2.39) for various values of N. Figure 2.7 shows that for N = 2, satisfying N > kr, good reconstruction is achieved, as shown by the sinusoidal behavior of the pressure. For N = kr = 1, some distortion is observed in the reconstructed sound pressure, while for N = 5, the reconstructed pressure is significantly different from the expected pressure. 2.4 Plane-Wave Composition A sound field composed of multiple plane waves can be represented as a summation over plane-wave terms, as in Eq. (2.37). When the sound field is composed of an infinite number of plane waves, or a continuum of plane waves, with directional amplitude density denoted by a(k,θ k,φ k ), then the sound pressure can be written as p(k, r, θ, φ) = = = 2π π n= m= n 2π π a(k,θ k,φ k )e i k r sin θ k dθ k dφ k 4πi n j n (kr)yn m (θ, φ) n= m= n a(k,θ k,φ k ) [ Y m n (θ k,φ k ) ] sin θk dθ k dφ k 4πi n a nm (k)j n (kr)yn m (θ, φ), (2.42) where a nm (k) is the spherical Fourier transform of a(k,θ k,φ k ). Comparing Eqs. (2.37) and (2.42), it is clear that for a sound field composed of a single unit-amplitude plane wave, the following holds:

13 2.4 Plane-Wave Composition 43 Fig. 2.7 Re{p(k, r,θ,φ)} due to a unit-amplitude plane wave with arrival direction (45, 45 ), evaluated using Eq. (2.39) and plotted on the surface of a sphere at kr = 1, for N = 5, 1, 2

14 44 2 Acoustical Background in which case, following Eq. (1.58), a nm (k) = [ Y m n (θ k,φ k ) ], (2.43) a(k,θ,φ)= δ(cos θ cos θ k )δ(φ φ k ). (2.44) When a sound field constructed from a composition of plane waves is evaluated at the surface of a sphere of radius r, it can be written in the spherical harmonics domain, following Eq. (2.42), as p nm (k, r) = 4πi n a nm (k)j n (kr). (2.45) This is a very useful result, relating directly the spherical harmonic coefficients of the plane-wave amplitude density to the spherical harmonic coefficients of the sound pressure. This is also the advantage of analyzing the sound pressure over the surface of a sphere the measured function p nm is in the same domain (spherical harmonics) as the function generating the sound field, a nm, thus facilitating a direct relation between to two. Equation (2.42) involves an infinite summation, but, similar to the case of a single plane wave, the infinite summation may be approximated in practice by a finite summation, leading to p(k, r, θ, φ) N n= m= n 4πi n a nm (k)j n (kr)yn m (θ, φ). (2.46) The properties derived for a finite-summation approximation of a sound field composed of a single plane wave also hold here, due to the similar dependence on the radial function j n (kr). Equations (2.4) and (2.45) suggest that complete information in a threedimensional space about a sound field composed of a single plane wave, or multiple plane waves, is available simply from the knowledge of the sound pressure at the surface of a single sphere. This is facilitated by the direct relation between the spherical harmonic coefficients of the sound pressure over a sphere, p nm (k, r), and the plane-wave amplitude density, a nm (k), composing the sound field in the entire space, as shown in Eq. (2.45). Given p(k, r,θ,φ) and having computed p nm (k, r) using the spherical Fourier transform, Eq. (1.41), the sound pressure at any other position in space, (r,θ,φ ), can be derived. First, the plane-wave amplitude density is computed by extracting a nm (k) through a division by 4πi n j n (kr) [see Eq. (2.45)] and then p nm (k, r ) is reconstructed by a multiplication with 4πi n j n (kr ) leading to p(k, r,θ,φ ) = n= m= n j n (kr ) j n (kr) p nm(k, r)y m n (θ,φ ). (2.47) This use of Eq. (2.47) is limited in practice by several factors. First, kr values corresponding to the zeros of the spherical Bessel functions lead to division by

15 2.4 Plane-Wave Composition 45 zero and a diverging quotient. Second, as discussed above, p nm (kr) has significant terms only up to order n = kr, whereas if r r, accurate reconstruction of the pressure at r will require terms up to order n = kr kr. Therefore, accurate calculation of p(k, r,θ,φ ) may require division by j n (kr), which may have low magnitude at n > kr, again leading to numerical instability. Furthermore, if the infinite summation in Eq. (2.47) is replaced by a finite summation of order N, as expressed in the following equation, the order-limited equation will be useful only in the range where both kr and kr are smaller than N: p(k, r,θ,φ ) N n= m= n j n (kr ) j n (kr) p nm(k, r)y m n (θ,φ ). (2.48) 2.5 Point Sources Real-world sources produce sound fields in their immediate vicinity with a behavior that makes it appropriate to model them as a simple point source (a monopole source) or a combination of these. Consider a point source located at r s = (r s,θ s,φ s ), producing unit-amplitude sound pressure at a distance of 1 m from the source. The source produces a spherical sound field, i.e. the pressure magnitude decays at a rate that is inversely proportional to the distance from the source, while the phase is constant as a function of θ and φ for a fixed distance from the source. The sound pressure at location r = (r,θ,φ) for this spherical radiation field can be written using a series of spherical harmonics as [23, 56] e ik r r s r r s = n= m= n 4π( i)kh (2) n (kr s)j n (kr) [ Y m n (θ s,φ s ) ] Y m n (θ, φ), r < r s, (2.49) where r =r and is the Euclidean norm. The condition r < r s means that the measurement point is nearer the origin relative to the point source. If a spherical measurement surface of radius r is considered, then the point source is assumed to be outside the measurement sphere. Note the similarity in this case between the sound field produced by the point-source and the plane-wave sound field, the latter described by Eq. (2.37), with plane-wave arrival direction replaced by the direction of the point source. Indeed, a point source positioned far from a measurement region will produce a sound field similar to a plane-wave sound field. The minus sign in the exponential e ik r r s guarantees that when combined with the time-dependent exponential, e iωt, the sound radiation is outwards from the point source. When the point source is nearer the origin relative to the measurement point, r and r s exchange places, such that

16 46 2 Acoustical Background e ik r r s r r s = n= m= n 4π( i)kh (2) n (kr)j n(kr s ) [ Y m n (θ s,φ s ) ] Y m n (θ, φ), r > r s. (2.5) Similarly, considering a spherical measurement surface of radius r, the point source is inside the measurement sphere. This equation is useful when analyzing sound radiation of sources by measuring the sound pressure at a surface surrounding the source. Note that in this case, the spherical harmonic coefficients of the sound pressure function measured at a sphere of radius r have radial dependence due (kr), rather than a spherical Bessel function dependence, j n (kr). The latter would be the case for a far point source or a plane wave. Although both spherical functions are solutions to the wave equation along r, the Hankel function has a singularity that is more suitable when describing a point source, as both produce infinite sound pressure at the singularity and source locations, respectively. The sound pressure at the surface of a sphere of radius r, p(k, r,θ,φ), due to a point source positioned at (r s,θ s,φ s ) can be described using the spherical harmonic coefficients by comparing Eq. (2.4) with Eqs. (2.49) and (2.5), leading to to the spherical Hankel function, h (2) n p nm (k, r) = 4π( i)kh (2) n (kr s)j n (kr) [ Y m n (θ s,φ s ) ], r < rs, (2.51) and p nm (k, r) = 4π( i)kh (2) n (kr)j n(kr s ) [ Y m n (θ s,φ s ) ], r > rs. (2.52) Equation (2.47), representing extrapolation of the sound pressure from a measurement sphere to other positions, can also be used for the case of a sound field generated by a point source, or by point sources, as long as the sources are outside the spheres of radii r and r. In the case where the sources are inside the spheres of radii r and r, j n (kr) and j n (kr ) in Eq. (2.47) should be replaced by h n (kr) and h n (kr ), respectively. Equation (2.49) can be used to describe the pressure around the origin at (r,θ,φ), for a point source that is positioned a significant distance away from the origin. Substituting the large argument approximation for the spherical Hankel function in this case, as in Eq. (2.34), the term h n (2)(kr n+1 e ikrs s) can be replaced by (i) kr s and, when substituted back into Eq. (2.51), the following approximation holds: p nm (k, r) e kr s r s 4πi n j n (kr) [ Y m n (θ s,φ s ) ], r < rs, kr s n(n + 1)/2. (2.53) The spherical harmonic coefficients of the sound pressure on a sphere of radius r are similar to the coefficients produced on the same sphere by a plane wave, as shown in Eq. (2.41), with (θ k,φ k ) = (θ s,φ s ) normalized by the term e krs r s representing the phase shift and attenuation due to the propagation from the point source to the origin. Furthermore, if we consider the sound pressure limited to a sphere of radius r and

17 2.5 Point Sources 47 approximately order limited to N = kr and assume that r s satisfies kr s > N(N +1)/2, then the sound pressure produced by the point source is approximately the same as the sound pressure produced by a plane wave with (θ k,φ k ) = (θ s,φ s ).Thisisa useful result, as it allows the sound pressure in a limited region in space, produced by a distant point source, to be approximated by the sound pressure produced by a plane wave and thus to inherit the properties of a plane-wave sound field. For a more detailed comparison between the sound field produced around the origin by a point source and by a plane wave, the reader is referred to [14]. 2.6 Sound Pressure Around a Rigid Sphere The sound pressure on the surface of a sphere in a free field due to plane waves and point sources has been analyzed in previous sections. In this section the pressure around a rigid sphere is derived. This is useful when measuring microphones are placed around a rigid sphere, which is often the case in practice, or when such a rigid sphere is employed to mimic a human head. The sound pressure around a rigid sphere is composed of the incident sound field, which is the sound field in free field without the rigid sphere, and the scattered sound field, which is the sound field that is scattered from the rigid sphere due to the incident field. The contribution of both fields to the sound pressure around a rigid sphere are formulated next. Consider a rigid sphere of radius r a. The sphere imposes a boundary condition on its surface of zero radial velocity: u r (k, r a,θ,φ)=, (2.54) because of the infinite impedance at the sphere boundary and the inability of the sound pressure to generate radial motion at this boundary. Acoustic velocity can be related to pressure through the equation of momentum conservation (or Euler equation) in spherical coordinates: iρ cku(k, r,θ,φ)= p(k, r,θ,φ), (2.55) where the gradient operator in spherical coordinates is given by p p r ˆr + 1 p r θ ˆθ + 1 p r sin θ φ ˆφ. (2.56) ρ is the air density in kilograms per cubic meter and ˆr, ˆθ, ˆφ are unit vectors, as shown in Fig. 2.8, with ˆr pointing in the direction of r, ˆθ is tangential to the surface of a sphere of radius r, pointing downwards along the longitude, and ˆφ is tangential to the surface of a sphere of radius r, pointing along the latitude. Substituting Eqs. (2.56) and (2.54) ineq.(2.55) with p = p i + p s and u r = u ri + u rs representing the total pressure and the total radial velocity, respectively, composed of the incident and scattered components, leads to

18 48 2 Acoustical Background Fig. 2.8 Spherical coordinate system showing coordinate directions [ pi (k, r,θ,φ)+ p s (k, r,θ,φ) ] r r=ra =. (2.57) The scattered pressure is now written as a spherical harmonics series as p s (k, r,θ,φ)= n= m= n c nm (k)h n (2) (kr)y n m (θ, φ). (2.58) Note the use of the spherical Hankel function, h n (2) (kr), as the scattered sound field originates from within the sphere of radius r, propagating outwards from the rigid sphere. The spherical Hankel function of the second kind is used; this is because it has terms of the form e ikr that, when combined with the time-dependent term, i.e. e i(ωt kr), suggest that the waves propagate in the positive ˆr direction, i.e. outwards from the rigid sphere. The incident sound pressure around the sphere can be written in the spherical harmonics domain as p i (k, r,θ,φ)= n= m= n a nm (k)4πi n j n (kr)yn m (θ, φ). (2.59) Note that a nm assumes an incident sound field composed of plane waves described in the notation previously used [see Eq. (2.42)]. However, a similar formulation also holds for sound fields due to point sources, as long as they are outside the sphere of radius r [see Eq. (2.49)]. Writing Eq. (2.57) in the spherical harmonics domain, by substituting Eq. (2.58) for the scattered pressure and Eq. (2.59) for the incident pressure, yields

19 2.6 Sound Pressure Around a Rigid Sphere 49 c nm (k) = a nm (k)4πi n j n (kr a) h (2) n (kr a ). (2.6) Substituting c nm in Eq. (2.58) and adding the incident field, Eq. (2.59), the total sound field around a rigid sphere is given by p(k, r, θ, φ) = n= m= n [ a nm (k)4πi n j n (kr) j n (kr ] a) h n (2) (kr a ) h(2) n (kr) Yn m (θ, φ). (2.61) By denoting [ b n (kr) = 4πi n j n (kr) j n (kr ] a) h (2) (kr a ) h(2) n (kr), (2.62) n the pressure outside the rigid sphere can be written in the spherical harmonics domain as p nm (k, r) = a nm (k)b n (kr). (2.63) Note the similarity to Eq. (2.45) with 4πi n j n (kr) replaced by b n (kr), now containing a scattering term. Also note that the explicit dependence of b n on r a has been omitted for notation simplicity. The behavior of the magnitude of b n, normalized by 4π, is presented in Fig Compared to Fig. 2.1, showing the magnitude of j n, function b n does not have zeros away from the origin. This important property is useful when a division by j n is replaced by a division by b n, such as in sound extrapolation [see Eq. (2.48)] or, generally, in array processing, as presented later in this book Fig. 2.9 Function b n (kr) /(4π) for a rigid sphere with r = r a,forn =,...,6

20 5 2 Acoustical Background Fig. 2.1 Function b n (kr) for a rigid sphere with r = r a and kr = 8, 16 Similar to the case of the pressure around a sphere in a free field, around a rigid sphere the magnitude of the spherical harmonic coefficients of the pressure due to a plane-wave sound field decreases for n > kr, as shown in Fig This figure is similar to Fig. 2.6, only here the functions are smoother for low values of n due to the absence of the zeros. Figure 2.11 shows the sound pressure, Re{p(k, r, θ, φ)}, around rigid spheres of radii r a = 1, 3, 1, due to a single unit-amplitude plane wave arriving from (θ k,φ k ) = (9, 2 ), with k = 1. The sound pressure was calculated using Eq. (2.61), with terms limited to order N = 32. Comparing Figs. 2.5 and 2.11, the effect of the sound pressure scattered from the rigid sphere is clear. For large radii, e.g. r a = 3, 1,

21 2.6 Sound Pressure Around a Rigid Sphere Fig Re{p(k, r,θ,φ)} for a unit-amplitude plane wave arriving from (θ k,φ k ) = (9, 2 ), with k = 1, plotted over the xy plane. Rigid spheres of radii r a = 1, 3, 1 m are positioned at the origin, also illustrated in the figure

22 52 2 Acoustical Background Fig Re{p(k, r,θ,φ)} due to a unit-amplitude plane wave with arrival direction (45, 45 ), evaluated using Eq. (2.61) withkr a = 1 and plotted on the surface of a rigid sphere the sound field around the rigid sphere is significantly altered by the scattered field, while for smaller radii, e.g. r a = 1, the change is minor. The relation between the radius of the rigid sphere and the magnitude of the scattered sound field can be studied analytically. The scattered sound field is dependent on the scattering term j n (kr a)/h n (2) (kr a ) in b n [see Eq. (2.62)]. For a small rigid sphere satisfying kr a 1, substituting the relation in Eq. (2.36) for the derivatives and using the small argument approximations in Eqs. (2.31) and (2.33), j n (kr a)/h n (2) (kr a ) is proportional to (kr a ) 2n+1 ; this term tends to zero as kr a, therefore leading to a negligible contribution from the scattered field. The sound pressure on the surface of a rigid sphere due to a plane-wave sound field is illustrated in Fig. 2.12, showing Re{p(k, r,θ,φ)} on the surface of a sphere of radius satisfying kr a = 1. The plane-wave arrival direction is (θ k,φ k ) = (45, 45 ), computed using Eq. (2.61) with terms up to order N = 32. The figure clearly shows that the magnitude of the sound pressure on the surface of the rigid sphere is highest at the location on the sphere near the arrival direction of the plane wave and attenuated along the propagation direction due to the effect of the rigid sphere. 2.7 Translations of Fields So far in this chapter the sound pressure has been presented relative to the origin of the spherical coordinate system. It may be useful to present the sound pressure in the spherical harmonics domain, relative to a translated spherical coordinate system. For example, the sound pressure around several spheres can be presented relative to a common origin. Other examples of translated sound fields represented in spherical

23 2.7 Translations of Fields 53 r Translated origin r r Origin Fig Translation of the origin to r harmonics have been investigated in recent publications [7, 39]. The aim of this section is therefore to provide an overview of the operation of translation of sound fields and of the effect of translation on the representation of the sound fields in spherical harmonics. Sound fields due to plane waves or distant point sources at (r,θ,φ)are described as a series of weighted j n (kr)yn m (θ, φ) terms, whereas sound fields due to point sources that are near the origin are described as a series of weighted h n (2)(kr)Y n m (θ, φ) terms [see Eq. (2.5)]. Consider a translation in the coordinate system from the origin to r = (r,θ,φ ), such that r = r + r, (2.64) as illustrated in Fig It may be useful to compute the coefficients of the sound field in the spherical harmonics domain with respect to the translated coordinates, relative to the original coefficients. Such a formulation can take different forms, depending on whether the original and the translated sound fields employ spherical Bessel or Hankel terms. The formulation of the translation therefore uses the following three transformations of terms; (i) from spherical Bessel functions to spherical Bessel functions [1], j n (kr)y m n (θ, φ) = n = n j n (kr )Yn m (θ,φ ) m = n j n (kr )Y m m n (θ,φ )C n nmn m, (2.65) n =

24 54 2 Acoustical Background (ii) from spherical Hankel functions to spherical Hankel functions, h (2) n (kr)y m n (θ, φ) = n = n h (2) n (kr )Yn m,φ ) m = n j n (kr )Y m m n (θ,φ )C n nmn m, r > r (2.66) n = and (iii) from spherical Hankel functions to spherical Bessel functions, where h (2) n (kr)y m n (θ, φ) = n = n j n (kr )Yn m,φ ) m = n h (2) n (kr )Y m m n (θ,φ )C n nmn m, r < r, (2.67) n = Cn nmn m = +n (2n + n) ( 1) m 1)(2n + 1)(2n + 1) 4πi(n 4π ( )( ) nn n n n n m m m m (2.68) ( ) j1 j and 2 j 3 is the Wigner 3-j symbol [1]. Equation (2.65) has been derived m 1 m 2 m 3 from the equation e i k r = e i k r e i k r by first substituting Eq. (2.37) for all terms and then multiplying by Yn m (θ k,φ k ) and integrating over the sphere with respect to (θ k,φ k ). Equations (2.66) and (2.67) can then be derived by exploring relationships between spherical Bessel and spherical Hankel functions [1]. We now consider the case where a sound field composed of multiple plane waves is measured around a spherical surface, r = (r,θ,φ), such that r is constant. In this case the function on the sphere can be represented by coefficients in the spherical harmonics domain, as in Eq. (2.45): p nm (k, r) = 4πi n a nm (k)j n (kr). (2.69) The coefficients a nm (k) provide information on the sound field and can be used to calculate the sound pressure at a position (r,θ,φ)relative to the origin. Now, keeping the same sound field, but shifting the origin of the coordinate system to r, we would like to calculate the sound pressure at position (r,θ,φ ) relative to the new origin, using a similar set of coefficients a nm (k). We would like to formulate a direct relation between a nm (k) and a nm (k). The sound pressure can be written using Eqs. (2.65) and (2.69)as

25 2.7 Translations of Fields 55 p(k, r, θ, φ) = = = n= m= n n= m= n 4πi n a nm (k)j n (kr)yn m (θ, φ) 4πi n a nm (k) n = n n = n j n (kr )Y m m n (θ,φ )C n nmn m 4πi n n = m = n [ n= m= n Therefore, the following holds: a n m (k) = n= m= n j n (kr )Y m n (θ,φ ) a nm (k) n = a nm (k) n = m = n j n (kr )Y m n (θ,φ ) j n (kr )Y m m n (θ,φ )C n nmn m. (2.7) ] j n (kr )Y m m n (θ,φ )C n nmn m, (2.71) which provides a relationship between the sound field coefficients in the original and in the translated coordinates. Similar relations can also be developed using Eqs. (2.66) and (2.67). Note that Cn nmn m is non-zero only for n n n n + n, and so if a nm is of finite order, each coefficient a n m can be calculated by a finite number of summations.

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