AND RAMANI DURAISWAMI

Transcription

1 MULTIPLE SCATTERING FROM N SPHERES USING MULTIPOLE REEXPANSION NAIL A. GUMEROV AND RAMANI DURAISWAMI Abstract. A semi-anaytica technique for the soution of probems of wave scattering from mutipe spheres is deveoped. This technique extensivey uses the theory for the transation and rotation of Hemhotz mutipoes that was deveoped in our earier work (Gumerov & Duraiswami, 21). Resuts are verified by comparison with commercia boundary eement software. The method deveoped is ikey to be very usefu in deveoping fast agorithms for many important probems, incuding those arising in simuations of composite media and mutiphase fow. Key words. mutipe scattering, Hemhotz equation, spherica harmonics, Mutipoe expansions 1. Introduction. Numerous practica probems of acoustic and eectromagnetic wave propagation require computation of the fied scattered by mutipe objects. Exampes incude scattering of acoustic waves by objects (e.g., the scattering of sound by humans and the environment), ight scattering by couds and environment, eectromagnetic waves in composite materias and the human body, pressure waves in disperse systems (aerosos, emusions, bubby iquids), etc. Fast and reiabe soution of forward scattering probems are especiay required for soving inverse probems arising, for exampe, in medica tomography, mine detection, and radar. Our interest is in the modeing the cues that arise due to scattering of sound and ight, and to use this information in simuating audio and video reaity (Duraiswami et a, 2). In many cases the scatterers are spheres, or can be modeed as such. Such modeing is convenient for parametrization of arge probems, since each sphere can be characterized by a few (say five) parameters, such as the three Cartesian coordinates of its center, its radius, and its impedance. This impedance wi in genera be a compex quantity, and characterizes the absorbing/refecting properties of the body/surface. For exampe, we are exporing the modeing of the human head and body using two spheres representing respectivey the head and the torso. In fuid mechanica probems, bubbes or dust partices can be assumed spherica (Gumerov et a (1988), Duraiswami and Prosperetti (1995)). Starting with the mutisphere representation, we can aso dea with the effect of perfecty refecting surfaces, such as was in fow channes, room was and the foor by repacing such surfaces with image spheres. For exampe, the acoustic fied in a rectanguar room with refecting was and a singe scattering sphere inside can be modeed by a set of image sources and spheres (Duraiswami et a, 21). However, anaytica soutions of the 3-D wave equation or the Hemhotz equation are avaiabe for ony very imited configurations of sphere geometries and boundary conditions. Usuay, such soutions are represented as infinite series of specia functions (such as in the simpest case of a singe sphere). In addition, ony in the cases of a singe sphere and two spheres is it possibe to introduce separabe systems of curviinear coordinates for which the sphere surface(s) coincide with a coordinate surface, eading to simpified treatment of boundary conditions. These cases cannot cover the case of, say three or more spheres, but are usefu as they provide an idea on the physics of scattering by spherica objects. In the genera case one can expect ony numerica, or approximate soutions. Numerica methods for soution of the Hemhotz equation in arbitrary domains, such as boundaryeement methods (BEM), finite-eement methods (FEM), or finite difference methods (FDM), are we known and extensivey used in research and commercia appications. Despite the reative advantages of these methods they a share a common deficiency reated to the necessity of discretization of either the boundary surfaces, or of the compete space. Discretization introduces a characteristic size (or ength scae) of the surface or spatia eement. For accurate and stabe computations the change of discretization ength must not affect the resuts of computations of the scattered fied. This eads to a requirement that this size theoreticay shoud be much smaer than the waveength λ, λ. Practicay this condition is < Bλ,where B is some constant smaer than 1. If computations are required for high frequencies (or short waves), this condition of stabiity/accuracy eads to very fine surface or spatia meshes. For exampe, for 2 khz sound in air in norma conditions the waveength is 1.7 cm. To compute scattering of such sound by atypicahumanheadofdiameterd =17cm the ength of one surface eement shoud be much smaer Perceptua Interfaces and Reaity Laboratory, Institute for Advanced Computer Studies, University of Maryand, Coege Park, MD Research supported by Nationa Science Foundation grant ITR This document is issued as University of Maryand Institute for Advanced Computer Studies Technica report UMIACS-TR-21-72, and aso as University of Maryand, Department of Computer Science Technica Report CS-TR

2 2 N.A. Gumerov & R. Duraiswami Incident Wave 6 Fig The mutipe scattering probem considered in this paper. than 1.7 cm, say 6 times smaer than 1.7 cm, i.e. 6 times smaer than the diameter of the head. The tota surface of the head is πd 2 and so the number of square eements shoud be of order π (D/ ) 2 12, this gives a 24 eement discretization of the head surface in case if trianguar eements are used in BEM. Such discretization require inversion of arge size eement inter-infuence matrices and are costy in terms of CPU time and memory, and cannot sti be reaized using even high end workstations. For simpified geometries, such as mutipe spheres, the scattering probem can be soved more efficienty using semi-anaytica techniques. In the present report we deveop such a method, which in some sense is anaytica since it is based on soutions in the form of infinite series. At the same time the method is numerica, since in the simpe form presented here, it requires inversion of a arge size matrix for determining coefficients in the series, to satisfy boundary conditions on mutipe spheres. The soution is based on decomposing the contributions of each scatterer to the tota fied, representation of each contribution in the form of series of spherica mutipoes of the Hemhotz equation, and reexpansion of each mutipoe near the center of each sphere to satisfy boundary conditions. This procedure produces infinite inear systems in the coefficients of the expansions. This system can be soved numericay by truncation of the series. We have deveoped software impementing this soution. We empoy recenty deveoped procedures of fast and stabe computation of genera mutipoe transation coefficients using recurrence reations (Gumerov & Duraiswami, 21). 1 We compare resuts of computations with numerica and anaytica soutions, and demonstrate the computationa efficiency of our method with commercia BEM software. The resuts showed that the deveoped method compares favoraby with commercia software in both accuracy and speed up of computations (in some cases by severa orders of magnitude). 2. Statement of the Probem. Consider the probem of sound scattering by N spheres with radii a 1,..., a N situated in an infinite 3 dimensiona space as shown in Fig The coordinates of the center of each sphere are denoted as r p =(x p,y p,z p), p=1,..., N. The scattering probem in the frequency domain is reduced to soution of the Hemhotz equation for compex potentia ψ (r), 2 ψ + k 2 ψ =, (2.1) 1 Since this reference wi be repeated many times, in the seque it wi be written as (GD21).

3 Mutipe Scattering from N Spheres 3 with the foowing impedance boundary conditions on the surface S p of the pth sphere: µ ψ n + iσ pψ =, p =1,..., N, (2.2) S p where k is the wavenumber and σ j are constants characterizing impedance of each sphere, and i = 1. In the particuar case of sound-hard surfaces (σ p =)we have the Neumann boundary conditions, ψ/ n Sp =, (2.3) and in the case of sound soft surfaces (σ p = ) we have the Dirichet boundary conditions, ψ Sp =. (2.4) Far from the region occupied by the spheres the compex potentia tends to the potentia of the incident wave ψ in (r): Usuay the potentia is represented in the form: ψ (r) r ψ in (r). (2.5) ψ (r) =ψ in (r)+ψ scat (r), (2.6) where ψ scat (r) is the potentia of the scattered fied. Far from the region occupied by spheres the scattered fied shoud satisfy the Sommerfed radiation condition: µ im r ψscat ikψ scat =. (2.7) r r 3. Soution Using Mutipoe Transation Reexpansion Decomposition of the Scattered Fied. Due to the inearity of the probem the scattered fied can be represented in the form ψ scat (r) = NX ψ p (r), (3.1) where ψ p (r) can be thought of as the fied scattered by the pth sphere. Each potentia ψ p (r) is a reguar outside the pth sphere and satisfies the Sommerfed radiation condition µ im r ψp r r ikψ p =, p =1,..., N. (3.2) 3.2. Mutipoe Expansion. Let us introduce N reference frames connected with the center of each sphere. In spherica poar coordinates r r p = r p =(r p, θ p, ϕ p ), soution of the Hemhotz equation that satisfy the radiation condition can be represented in the form ψ p (r) = nx n= m= n Here A (p)m n are coefficients, S m n (r) is a mutipoe of order n and degree m : p=1 A (p)m n S m n (r p ), p =1,..., N. (3.3) S m n (r p )=h n (kr p )Y m n (θ p, ϕ p ), p =1,..., N, (3.4) h n (kr) are spherica Hanke functions of the 1st kind that satisfy the Sommerfed condition, and Yn m (θ, φ) are orthonorma spherica harmonics, which aso can be represented in the form s Yn m (θ, ϕ) =( 1) m 2n +1(n m )! 4π (n + m )! P n m (cos θ)e imϕ, (3.5) n =, 1, 2,..., m = n,..., n,

4 4 N.A. Gumerov & R. Duraiswami M r q r q r p r pq p r q r p O Fig Notation denoting the different reference frames used in the mutipoe re-expansion technique. where Pn m (µ) are the associated Legendre functions. The probem now is to determine coefficients A (p)m n so that thecompetepotentia NX nx ψ (r) =ψ in (r)+ A (p)m n Sn m (r p ) (3.6) p=1 n= m= n satisfies a the boundary conditions on the surface of each sphere Mutipoe Reexpansion. To sove this probem et us consider the qth sphere. Near the center of this sphere, r = r q, a the potentias ψ p (r), are reguar for p 6= q. EachmutipoeS n m (r p ),p6= q can be then re-expanded into a series near this center, r q 6 r p r q as foows: S m n (r p)= X = s= (S R) sm n r pq R s (r q ), p,q =1,..., N, p 6= q. (3.7) Here Rn m (r q) are reguar eementary soutions of the Hemhotz equation in spherica coordinates connected with the qth sphere: Rn m (r q )=j n (kr q )Yn m (θ q, ϕ q ), p =1,..., N, (3.8) where j n (kr) are spherica Besse functions of the first kind. The coefficients (S R) sm n r pq are the transation reexpansion coefficients, and depend on the reative ocations of the pth and qth spheres, r pq. Since r = r p + r p = r q + r q,wehave r p = r q + r pq, r pq = r q r p = r p r q, (3.9) where r pq is the vector directed from the center of the pth sphere to the center of the qth sphere. Detaied investigation of mutipoe reexpansion coefficients, their computation, their properties, and methods for efficient evauation are considered in GD21. Near this center of expansion the incident fied can be aso represented using a simiar series, the radius of convergence for which is not smaer than the radius of the qth sphere: X ψ in (r) = C (in)s r q R s (r q ). (3.1) = s=

5 Mutipe Scattering from N Spheres 5 Substituting oca expansions (3.7) and (3.1) into (3.6) we obtain the foowing representation of the fied near r = r q ψ (r) = X = s= C (in)s r q R s (r q )+ nx n= m= n A (q)m n S m n (r q )+ NX nx p=1 n= m= n p6=q A (p)m n X = s= (S R) sm n r pq R s (r q ). Let us change the order of summation in the atter term and substitute expressions for S m n and R m n,(3.7) and (3.8). This expression then can be rewritten as ψ (r) = X h = s= B (q)s i j (kr q )+A (q)s h (kr q ) Y s (θ q, ϕ q ), (3.11) B (q)s (r 1,..., X N r N )=C(in)s r q + nx p=1 n= m= n p6=q (S R) sm n r pq A (p)m n. (3.12) 3.4. Boundary Conditions. From these equations we have the foowing reations for the boundary vaues of ψ and its norma derivative on the surface of the qth sphere ψ Sq = ψ n Sq = k X h = s= X h = s= B (q)s B (q)s i j (ka q )+A (q)s h (ka q ) Satisfying boundary condition (2.2) on the surface of the qth sphere, we have X n = s= B (q)s Y s (θ q, ϕ q ), (3.13) i j (ka q )+A (q)s h (ka q ) Y s (θ q, ϕ q ), (3.14) o [kj (ka q )+iσ q j (ka q )] + A (q)s [kh (ka q )+iσ q h (ka q )] Y s (θ q, ϕ q )=. (3.15) Orthogonaity of the surface harmonics yieds: B (q)s [kj (ka q )+iσ q j (ka q )] + A (q)s [kh (ka q )+iσ q h (ka q )]=, =, 1,..., s =,...,. (3.16) Note that the boundary vaues of ψ Sq and ψ/ n Sq can be expressed in terms of coefficients A (q)s, since B (q)s = A (q)s and formuae (3.13) and (3.14) yied ψ Sq = ψ n Sq = k X = s= X = s= kh (ka q)+iσ q h (ka q ) kj (ka, =, 1,..., s =,...,. (3.17) q)+iσ q j (ka q ) h (ka q ) j (ka q ) kh (ka q)+iσ q h (ka q ) kj (ka q)+iσ q j (ka q ) h (ka q ) j (ka q ) kh (ka q)+iσ q h (ka q ) kj (ka q)+iσ q j (ka q ) A (q)s Y s (θ q, ϕ q ), (3.18) A (q)s Y s (θ q, ϕ q ). (3.19) These reations can be aso rewritten in a compact form using the Wronskian for the spherica Besse functions, W {j (ka),h (ka)} = j (ka)h (ka) j (ka)h (ka) =i(ka) 2 (3.2)

6 6 N.A. Gumerov & R. Duraiswami as ψ Sq = 1 X A (q)s Y s(θ q, ϕ q ) ika 2 q kj = s= (ka q)+iσ q j (ka q ), (3.21) ψ = σ X q A (q)s Y s(θ q, ϕ q ) n ka 2 q kj (ka q)+iσ q j (ka q ) = iσ q ψ Sq. (3.22) Sq = s= For the particuar case of a sound-hard spheres (σ q =)thisgives ψ Sq = 1 ik 2 a 2 q X = s= A (q)s Y s(θ q, ϕ q ) j (ka, q) ψ =, (3.23) n Sq whie for sound-soft (σ q = ) wehave ψ Sq =, ψ = i n Sq ka 2 q X = s= A (q)s Y s(θ q, ϕ q ). (3.24) j (ka q ) 3.5. Matrix Representation. To determine the boundary vaues of the potentia and/or its norma derivative and obtain a spatia distribution according (3.6) we need to determine the coefficients A (q)s in equations (3.16) and (3.12), which are vaid for any sphere, q =1,..., N. These equations form a inear system that may be represented in standard matrix-vector form. This can beaccompishedindifferent ways according to the probem. First, we note that coefficients of expansions to spherica harmonics, such as A m n,n =, 1, 2,...; m = n,..., n, can be aigned into one coumn vector, e.g. A = A,A 1 1,A 1,A 1 1,A 2 2,A 1 2,A 2,A 1 2,A 2 2,... T, (3.25) where the superscript T denotes the transpose. reated to coefficients A m n by In this representation the eements of the vector A are A t = A m n, t =(n +1) 2 (n m), n =, 1, 2,...; m = n,..., n; t =1, 2,... (3.26) The same reduction in dimension can be appied to coefficients of reexpansion, (S R) sm n. Instead of a 4 dimensiona matrix the coefficients can be packed in a 2 dimensiona matrix as (S R) (S R) 1 1 (S R) 1 (S R) 1 1 (S R) (S R) 1 1 (S R) (S R) 1 11 (S R) (S R) (S R) = (S R) 1 (S R) 1 11 (S R) 11 (S R) 1 11 (S R) (S R) 1 1 (S R) (S R) 1 11 (S R) (S R) , (3.27) (S R) 2 2 (S R) (S R) 2 21 (S R) (S R) with the foowing correspondence of the matrix eements and coefficients: (S R) rt =(S R) sm n, r =( +1)2 ( s), t =(n +1) 2 (n m) (3.28), n =, 1, 2,...; m = n,..., n; s =,...,. Using this representation we introduce the foowing vectors and matrices n o T n o T n A (q) =, D (q) =, L (qp) = A (q) t q =1,..., N, p =1,..., N. D (q) t L (qp) rt o, (3.29)

7 where A (q) t Mutipe Scattering from N Spheres 7 = A (q)m n, (3.3) D (q) t = kj n (ka q )+iσ q j n (ka q ) kh n (ka q )+iσ q h n (ka q ) C(in)m n r q, L (qp) rt = kj (ka q)+iσ q j (ka q ) kh (ka q)+iσ q h (ka q ) (S R)sm n r pq, for p 6= q, L (qq) rt = δ rt, t =(n +1) 2 (n m), n =, 1, 2,...; m = n,..., n; r =( +1) 2 ( s), =, 1, 2,...; s =,...,. q =1,..., N, p =1,..., N, where δ rt is the Kronecker deta, δ rt =for r 6= t and δ rr =1. Equations (3.16) and (3.12) then can be represented in the form NX L (qp) A (p) = D (q), q =1,..., N. (3.31) p=1 This system of equations can be represented as a singe equation of the form where the tota matrices and vectors can be formed as L (11) L (12)... L (1N) L = L (21) L (22)... L (2N) , A = L (N1) L (N2)... L (NN) LA = D, (3.32) A (1) A (2)... A (N), D = D (1) D (2)... D (N). (3.33) This inear system can be soved numericay using standard routines, such as LU-decomposition. An important issue is the truncation of the infinite series and corresponding truncation of the associated matrices. A first soution is that we seect a number M of modes to retain for each expansion. This number is seected via a heuristic based on the magnitude of the smaest retained term. In this case n =, 1,..., M, m = n,..., n, then the ength of each vector A (p) and D (p) wi be (M +1) 2, and the size of each sub-matrix L (qp) wi be (M +1) 2 (M +1) 2, the size of the tota vectors A and D wi be N(M +1) 2 and the size of the tota matrix L wi be N(M +1) 2 N(M +1) 2. Severa areas of research are ongoing to improve the soution procedure. We are currenty investigating the use of fast mutipoe methods to sove these equations iterativey. Aso, we are trying to put the heuristics used for truncation of the series on a more firm basis Computation of Mutipoe Reexpansion Coefficients. The (S R)-mutipoe transation coefficients can be computed in different ways incuding via numerica evauation of integra representations, or using the Cebsch-Gordan or 3-j Wigner symbos (e.g. see Epton & Dembart, 1994, Koc et a., 1999). For fast, stabe, exact and efficient computations of the entire matrix of the reexpansion coefficients we used a method based on recurrence reations, which we have deveoped. We refer the reader to GD21 for proofs and detais, and ony provide necessary reations and initia vaues for using the recurrence procedures here. A the (S R) sm n r pq transation reexpansion coefficients can be computed in the foowing way. First, we compute so-caed sectoria coefficients of type (S R) sm m and (S R)sm s n using the foowing reations: b m 1 m+1 (S R) s,m+1 (S R) s 1,m 1,m bs 1 +1 (S R)s 1,m +1,m, (3.34) =, 1,... s=,...,, m =, 1, 2,...,m+1 = b s b m 1 m+1 (S R) s, m 1,m+1 = b s (S R) s+1, m 1,m b s 1 +1 (S R) s+1, m +1,m, (3.35) =, 1,... s =,...,, m =, 1, 2,...

8 8 N.A. Gumerov & R. Duraiswami where b m n = q (n m 1)(n m) q (n m 1)(n m) (2n 1)(2n+1) (2n 1)(2n+1), 6 m 6 n,, n 6 m<, m >n,, (3.36) and the recurrence process starts with (S R) s Due to the symmetry reation r pq = p (4π)( 1) S s (S R) m, s m r pq, (S R) m n r pq = p (4π)S m n r pq. (3.37) =( 1) +m (S R) sm m, (3.38) =, 1, 2,..., s =,...,, m = n,..., n, a of the sectoria coefficients (S R) sm s n can be obtained from the coefficients (S R)sm m. Once the sectoria coefficients are computed a other coefficients can be derived from them using the foowing recurrence reation, which does not change the degrees s, m of the reexpansion coefficients: where a m n 1 (S R) sm,n 1 am n (S R) sm,n+1 = as (S R) sm +1,n as 1 (S R) sm 1,n, (3.39), n =, 1,... s =,...,, m = n,..., n, a m n = ( q (n+1+ m )(n+1 m ) (2n+1)(2n+3), n > m,, m >n.. (3.4) Due to symmetry (S R) sm n =( 1)n+ (S R) m, s n, (3.41), n =, 1,... s=,...,, m = n,..., n. ony those coefficients with > n need be computed using recurrence reations. Aso, the (S R) coefficients for any pair of spheres p an q need be computed ony for the vector r pq, since for the opposite directed vector we have: (S R) sm n r qp =( 1) +n (S R) sm n r pq,,n =, 1,..., m = n,..., n. (3.42) 4. Genera Computationa Agorithm. A fow chart of the computationa agorithm is shown in Fig Software based on this agorithm was deveoped and entited MutisphereHemhotz. Resuts of tests using this software are discussed beow. 5. Coaxia Spheres. The case of two spheres is interesting, since on one hand the scattered fieds due to the two spheres interact with each other (mutipe scattering), whie on the other the interaction is sti simpe enough that it can be investigated in more and understood more intuitivey than the genera case of N spheres. Additionay, in this case, the computation of the reexpansion matrices can be simpified by proper seection of the reference frames. Indeed, for two spheres we can introduce a reference frame which has its z axis directed from the center of one sphere to the center of the other sphere. Since the reexpansion coefficients depend ony on the reative ocations of the spheres, for this frame orientation, there wi be no anguar dependence of these coefficients. The same statement hods for the case when there are N spheres arranged aong a ine, taken to be the z axis. In these particuar cases, the genera reexpansion formua (3.7) simpifies to S m n (r p )= = m (S R) mm n r pq R m (r q ), p =1,..., N, p 6= q. (5.1)

9 Mutipe Scattering from N Spheres 9 An Agorihm for Soution of the Genera Probem for N spheres Specify Parameters of N spheres (ocations, sizes, and impedances) Specify the Incident Fied (ocation and type of acoustic sources) Specify Maximum Truncation number M max. Form Right Hand Side Vector D Compute a Surface Harmonics, Y, for n =,, M max Specify Wavenumber k and series/matrices truncation number M(k). Compute a Hanke and Besse Functions Find vector A by Soving Linear System LA = D Compute Potentia/Norma Derivatives for Specified Locations on Sphere Surfaces Compute Potentia for Specified Locations in Space Compute a Reexpansion Coefficients, τ Postprocess/Output Resuts Form Genera System Matrix L Another Incident Fied? yes LU-decompose L and store yes Another Wavenumber? no End Fig Fow chart of the genera computationa agorithm. The coefficients (S R) m n r pq =(S R) mm n r pq,,n =, 1,..., m = n,..., n, (5.2) satisfy genera recurrence reations and can be computed using the genera agorithm we have deveoped. However, the simper reations that take advantage of the co-axiaity of the spheres resut in faster computation, and provide substantiay ower dimensiona matrices. The recurrence reations and computationa agorithm of the coaxia reexpansion coefficients can be found in our report (GD21). Note that the sign of coefficients (S R) m n r pq depends on the direction of the vector r pq.tobedefinite we seect by convention that notation rpq corresponds to r pq and r qp corresponds to r qp = r pq. Since (S R) mm n r pq =( 1) +n (S R) mm n r pq =( 1) +n (S R) mm n r qp (see GD21), we wi have: (S R) m n r pq =( 1) +n (S R) m n r qp,,n =, 1,..., m = n,..., n. (5.3) 5.1. Matrix Representation. According (5.1) harmonics of each degree m can be considered independenty. Equations (3.12) and (3.16) can be rewritten in the form: kj (ka q)+iσ q j (ka q ) kh (ka q)+iσ q h (ka q ) NX p=1 n= m p6=q (S R) m n r pq A (p)m n + A (q)m = D (q)m, (5.4) m =, ±1, ±2,..., = m, m +1,..., q =1,...,N, where D (q)m = kj (ka q)+iσ q j (ka q ) kh (ka q)+iσ q h (ka q ) C(in)m r q, m =, ±1, ±2,..., = m, m +1,..., q =1,..., N. (5.5)

10 1 N.A. Gumerov & R. Duraiswami This inear system can be represented in the foowing form NX L (qp)m A (p)m = D (q)m, m =, ±1, ±2,..., q =1,..., N, (5.6) p=1 where the vectors A (q)m and D (q)m and matrices L (qp)m are packed as foows n o T n o T n o A (q) =, D (q)m =, L (qp)m =, (5.7) A (q)m n D (q)m n L (qp)m n q =1,..., N, p =1,..., N, m =, ±1, ±2,...,, n = m, m +1,... with the individua matrix eements given by L (qp)m n = kj (ka q)+iσ q j (ka q ) kh (ka q)+iσ q h (ka q ) (S R)m n r pq, for p 6= q, (5.8) L (qq)m n = δ n. Since a equations can be considered separatey for each m, the inear system (5.6) can be written as where L m = L (11)m L (12)m... L (1N)m L (21)m L (22)m... L (2N)m L (N1)m L (N2)m... L (NN)m L m A m = D m, m =, ±1, ±2,..., (5.9), Am = A (1)m A (2)m... A (N)m, Dm = D (1)m D (2)m... D (N)m. (5.1) As in the genera case considered above, the infinite series and matrices shoud be truncated for numerica computation. If we imit ourseves to the first M modes in each expansion of spherica harmonics, so that m =, ±1,..., ±M, n = m, m +1,..., M, then the ength of each vector A (p)m and D (p)m is M +1 m, the dimensions of each matrix L (qp)m is (M +1 m ) (M +1 m ), the size of the tota vectors A m and D m are N(M +1 m ) and the size of the tota matrix L m is N(M +1 m ) N(M +1 m ). The probem then is reduced to soution of 2M +1 independent inear systems for each m. Note that the coaxia, or diagona, transation coefficients (S R) m n r pq are symmetrica with respect to the sign of the degree m, (S R) m n r pq =(S R) m n r pq (see GD21). Therefore the matrices L m are aso symmetrica (see (5.8)) L m = L m, m =, 1, 2,..., (5.11) and can be computed ony for non-negative m. Atthesametimetheright-handsidevectorD m, generay speaking, does not coincide with D m, so that the soution A m can be different from A m. Let us compare the number of operations required for determination of a expansion coefficients A n (q)m, m =, ±1,...,, ±M, n = m, m +1,..., M, q =1,..., N, using the genera agorithm and using the agorithm for coaxia spheres. Assuming that a standard sover requires CK 3 operations to sove a inear system with matrix K K, where C is some constant, we can find soution using genera agorithm for N (genera) operations = CN3 (M +1) 6 (5.12) operations. Using the agorithm for coaxia spheres we wi spend CN 3 (M +1 m ) 3 operations to obtain A (q)m n for each m =, ±1,...,, ±M. The tota number of operations wi be therefore # MX MX N (coaxia) operations = CN3 (M +1 m ) 3 = CN "(M 3 +1) 3 +2 m 3 (5.13) m= M m=1 = CN 3 (M +1) M 2 (M +1) 2 = 1 2 CN3 h (M +1) 4 +(M +1) 2i.

11 Therefore, for M À 1, we have Mutipe Scattering from N Spheres 11 N (genera) operations N (coaxia) 2M 2. (5.14) operations which shows much higher efficiency of the agorithm for coaxia spheres. Note aso that in the case of coaxia spheres the number of the mutipoe reexpansion coefficients that need to be computed for each pair p 6= q requires O(M 3 ) operations, whie in genera case such computations can be performed in O(M 4 ) operations (see GD21). These numbers are smaer than the eading order term in the compexity, that required for soution of inear equations, and thus evauations (5.12)-(5.14) provide a comparison between two methods Rotation of the Reference Frame. The above expressions for coaxia spheres assume that the z axis coincides with the direction from the center of one of the spheres to the center of some other sphere. If coordinates of spheres are specified in origina reference frame, with axis z oriented arbitrariy with respect to the ine connected the sphere centers we can rotate them so that the new reference frame is convenient for use with the coaxia agorithm. Therefore we consider two reference frames: the origina reference frame and the reference frame which axis z is directed say from sphere 1 to sphere 2. Denote the coordinates of an arbitrary point in the origina reference frame as (x, y, z) and the coordinates of the same point in the rotated reference frame, with the same origin, as (bx, by,bz). Denote the unit coordinate vectors of the origina coordinates as (i x, i x, i z ) and the unit vectors of the rotated reference frame as (i bx, i by, i bz ). By definition, we have the foowing representations for an arbitrary vector r : r =xi x + yi y + zi z = bxi bx + byi by + bzi bz. (5.15) Coordinates (x, y, z) and(bx, by, bz) are connected with the rotation matrix Q : bx by = Q x y, x y = Q bx by, (5.16) bz z z bz where By definition Q = i bx i x i bx i y i bx i z i by i x i by i y i by i z i bz i x i bz i y i bz i z, Q = Q 1 = i bx i x i by i x i bz i x i bx i y i by i y i bz i y i bx i z i by i z i bz i z. (5.17) i bz = r 12 r 12 = r 12 = e x i x + e y i y + e z i z, r 12 e 2 x + e2 y + e2 z =1, (5.18) where e x,e y, and e z are the direction cosines of i bz, and e x = r 12 i x, e y = r 12 i y, e z = r 12 i z. (5.19) r 12 r 12 r 12 As it is shown in our previous report (GD21) in this case the rotation matrix can be computed as a composition of two rotations: e q(e y e x q(e 2 cos φ sin φ x +e 2 2 y) x +e 2 y) Q = sin φ cos φ q q(e 1 e ze x e ze y e q(e 2 2 x +e2 2 x + ey 2 y) x +e2 y), (5.2) e x e y e z where φ is an arbitrary ange, meaning that any rotation around the i bz preserves i bz and satisfies our requirement for a convenient reference frame. Particuary, for φ =, the first mutipier in the right hand side of (5.2) is the identity matrix. Obviousy a reations for coaxia spheres and expansion coefficients A (p)m n shoud be then treated in the rotated reference frame, which formay means addition of hats over a coordinates in the expressions derived. A spherica harmonics and mutipoes represent soutions in the rotated reference frame and the incident fied expansion coefficients C (in)m shoud aso be found in the (i bx, i by, i bz ) system of coordinates.

12 12 N.A. Gumerov & R. Duraiswami 5.3. Computation of Coaxia Reexpansion Coefficients. Due to the symmetry reations (see (3.42) and (5.2)) and (S R) m n =(S R) m n, m =, 1, 2,... (5.21) the coaxia coefficients (S R) m n r pq can be computed ony for > n > m >. The process of fiing of the matrix {(S R) m n } can be performed efficienty using recurrence reations that first fi the ayers with respect to the orders and n foowed by advancement with respect to the degree m. If such a fiing procedure is seected then the first step is fiing of the ayer m =. Theinitiavauedependsontheorientationofr pq vector reative to the axis i z (or i bz if rotation is performed), and is given by (S R) p r pq =(S R) r pq = (4π)( 1) S p r pq = (2 +1)h (krpq), (5.22) where Ã! ½ ² pq = r pq i z ( 1) r =, for r pq i z = r pq 1, for r pq pq i z = r. (5.23) pq For advancement with respect to m it is convenient to use (5.4) for n = m, b m 1 m+1 (S R) m+1,m+1 = b m 1 (S R) m 1,m bm +1 (S R) m +1,m, = m +1,m+2,..., (5.24) with b s given by (3.36) and obtain other (S R) m+1 n using (3.39). For the case the axis is directed in the opposite direction, the coefficients wi have aternating signs. This is easy to estabish using the definition (5.2) and the symmetries (3.42), (5.3), and (5.21) Computationa Agorithm. The computationa agorithm for the case of coaxia spheres amost repeats the genera agorithm. Pecuiarities incude rotation of the reference frame and the packing of the system matrix to reduce the memory compexity. Note that since computations are performed in the reference frame (i bx, i by, i bz ), a vectors and surface harmonics are represented in that frame. With respect to the computationa reference frame, vectors such as r shoud be treated by appying the rotation matrix Q to them. To store the matrix (and its LU-decomposition) in an optima way we use a muti-index packing in a one-dimensiona array. The agorithm was reaized in software entited MutisphereHemhotzCoaxia. The code was vaidated by comparisons with the genera code MutisphereHemhotz. Since both codes produced the same resuts in the case of coaxia spheres, no more discussions on comparisons of the resuts wi provided in this report. As expected MutisphereHemhotzCoaxia showed better performance compared to MutisphereHemhotz to achieve the same accuracy (the same truncation number) in terms of CPU time and memory. C (in)s 6. Typica Incident Fieds. Beow we consider expicit expressions for mutipoe expansion coefficients r q (see (3.1)) for severa typica incident fieds encountered in practica probems Monopoe Source. Inthecaseofamonopoesourceocatedatr = r source, the incident fied corresponds to the fundamenta soution of the Hemhotz equation ψ in (r) =QG k (r r source )=Q eik r rsource 4π r r source, (6.1) where Q is the source intensity (compex, if the phase Φ is not zero) Q = Q e iφ. (6.2) Expansion of this function near the center of the qth sphere r = r q (see aso Fig. 2) can be found esewhere (e.g., Morse & Feshbach, 1953) nx ψ in (r) =Qik Sn m rsource r q R m n (r q ), r q 6 r source rq. (6.3) n= m= n Comparing (3.1) and (6.3) we obtain C (in)m n r q = QikS m n rsource r q. (6.4)

13 Mutipe Scattering from N Spheres Mutipe Monopoe Sources. This type of incident fied is encountered not ony when the fied is generated by severa rea sources, but aso when additiona sources appear as images, as when panes of symmetry bound the domain. For exampe if we have a singe sphere near an infinite rigid wa, and where the incident fied is generated by a singe monopoe source, the probem in the haf-space is equivaent to the probem in the whoe space, but for two spheres and two sources. The image sphere and source are symmetrica to the rea sphere and source with respect to this pane. More generay, the incident fied generated by N s sources is XN s ³ ψ in (r) = Q α G k r r (s), (6.5) α=1 where Q α and r (s) α are respectivey the compex intensities and ocation of the αth source (α =1,...,N s ). Using the expansion for a singe source (6.3) near the center of the qth sphere r = r q, and superposing the fied due to a sources, we have ψ in (r) =ik nx XN s n= m= n α=1 Comparing (3.1) and (6.3) we obtain ³ Q α Sn m r (s) α r q R m n (r q ), r q 6 C n (in)m XN s r q = ik α=1 α r (s) α r q. (6.6) ³ Q α Sn m r (s) α r q. (6.7) 6.3. Pane Wave. The case of sound scattering by a rigid sphere paced in a pane wave was considered by Lord Rayeigh in the 19th century (Strutt, 194, 1945). We incude this case for generaity and to provide some formuae for arbitrary ocations of the origin of the reference frame. The incident fied of the pane wave is described by where ψ in (r) =Qe ik r, (6.8) k =(k x,k y,k z ), k 2 x + k 2 y + k 2 z = k 2, (6.9) isthewavevector,andq is the compex ampitude of the wave (6.2). Expansion of the fied in spherica harmonics near the center of the qth sphere r = r q can be obtained from the soution for monopoe source, Equations (6.1) and (6.3), where we use an asymptotic expansion of the spherica Hanke function at arge distances between the source and the sphere: e ψ in (r) =4πQ r ik r r " # source nx source =4πQ r source ik S m n rsource r q R m 4π r r source n (r q ) r source n= m= n r source (6.1) " # =4πQ nx X ikr source n= m= n =4πQe ikr q,source =4πQe ikr q,source nx n= m= n nx n= m= n h n (kr q,source ) Y m n (θ q,source, ϕ q,source )R m n (r q ) i n Y m n (θ q,source, ϕ q,source )R m n (r q ) ( 1) n i n Yn m (θ k, ϕ k )Rn m (r q ). r source Here r q,source = r source r q, and the anges θ k = π θ q,source and ϕ k = π +ϕ q,source determine the direction of the wave from the infinitey ocated source, and therefore they can be obtained from the known wave

14 14 N.A. Gumerov & R. Duraiswami vector (6.9) as sin θ k cos ϕ k = k x /k, (6.11) sin θ k sin ϕ k = k y /k, cos θ k = k z /k. q Now we note that r = r q + r q (see Fig. 2). If we put r q = in (6.1) and use R s () = 1 4π δ δ s,wefind that and, more generay, ψ in (r) =4πψ in r q X On the other hand, using definition (6.8) ψ in r q = Qe ikr q,source, (6.12) nx n= m= n ( 1) n i n Yn m (θ k, ϕ k )Rn m (r q ). (6.13) ψ in r q = Qe ik r q. (6.14) Comparing (3.1) with (6.13) and (6.14), we obtain the foowing expression for C n (in)m (r 1) as C n (in)m r q =4πQe ik r q ( 1) n i n Yn m (θ k, ϕ k ). (6.15) This resut can be easiy generaized to the case where the incident fied is obtained from the superposition of N w pane waves: XN w ψ in (r) = Q α e ikα r, k α =(k α,x,k α,y,k α,z ), kα,x 2 + kα,y 2 + kα,z 2 = k 2. (6.16) α=1 with intensities Q α and incident anges θ (α) k (6.11)). In this case we have, ϕ(α) k C n (in)m XN w r q =4π( 1) n i n Q α e ik α r q Y m n, corresponding to the wave vectors k α (α =1,...,N w, see α=1 (θ (α) k, ϕ(α) k ). (6.17) 6.4. Mutipoe Sources. The case of mutipoe sources is a practica one to consider, since rea sources of sound (oudspeakers, turbuent vortices, etc.) usuay have not ony monopoe, but aso dipoe, and quadrupoe components. More generay, for the case of a mutipoe source of order n and degree m we have ψ in (r) =QS m n (r r source ), (6.18) where r source is the ocation of the source, and Q is its compex ampitude. In this case, for determination of the expansion of the incident fied near the center of the qth sphere we can use resuts obtained for mutipoe reexpansion (3.7) S m n (r r source )= Coefficients C (in)s r q then simpy become = s= X (S R) sm r q r source R s (r q ). (6.19) n C (in)s r q = Q(S R) sm n r q r source. (6.2)

15 Mutipe Scattering from N Spheres 15 This resut can be generaized to the incident fied generated by N s mutipoe sources, where each source has its own intensity Q α, ocation r (s) α,ordern α, and degree m α (α =1,..., N s )as XN s ψ in (r) = α=1 ³ Q α Sn mα α For such a fied, the coefficients C (in)s r q can be computed as foows α=1 r r (s), (6.21) α C (in)s XN s ³ r q = Q α (S R) smα n α r q r (s) α. (6.22) Note that this formua covers many practica cases, where the fied is generated by different types of sources and their images due to the presence of panes or other refecting surfaces, if a representation of the type (6.21) is avaiabe for the particuar geometry of the probem. 7. Exampe Probems. We appy our method to severa exampe probems. First, we compare the semi-anaytica soution we obtain with the purey numerica soutions obtained with the aid of the boundary eement method (BEM) to determine the accuracy of soution, and identify probems that wi be addressed in subsequent studies Singe Sphere. For the case of singe sphere there is no need to sove a inear system to account for the infuence of neighboring spheres. However, this case is important from a practica point of view. We incude this case for demonstration of the soution, comparison with the BEM, error evauations, and to provide anaytica soution for the probem of the fied generated by an arbitrary mutipoe in the presence of a sphere. In the case of singe sphere (N =1,q =1)from (3.3) - (3.33) we have L (1) = I, A (1) = D (1), or A (1)m n = kj n (ka)+iσj n (ka) kh n (ka)+iσh n (ka) C(in)m n (r 1), (7.1) Here and beow in the case of a singe sphere we wi drop subscript 1 for σ and a, whie keeping them for the coordinates. Substituting this expression into (3.21) and (3.22), we obtain expressions for the potentia and its norma derivative on the sphere surface as ψ S1 = 1 ika 2 ψ n S1 = σ ka 2 X = s= X = s= C (in)s (r 1) Y s(θ 1, ϕ 1 ) kh (ka)+iσh (ka), (7.2) C (in)s (r 1) Y s(θ 1, ϕ 1 ) kh (ka)+iσh (ka) = iσ ψ S 1. (7.3) The coefficients C n (in)m (r 1) are determined using (3.1) and depend on the incident fied. We consider the foowing particuar cases for the incident fied Monopoe Source. Asimpification of genera formua for C n (in)m (6.4) is possibe, since the probem for monopoe source and a singe sphere is axisymmetric reative to the axis connecting the center of the sphere and the ocation of the source. In this case seecting the axis z 1 to be the axis of symmetry, we have r r 2n +1 2n +1 C n (in)m = Qikδ m h n (k r source r 1 ) Yn (, ) = Qikδ m 4π h n (kd) =Qikδ m 4π h n (kd), (7.4) where d is the distance between the source and the center of the sphere. In this most simpified case of the reference frame seection, expressions for the surface vaues of the potentia and its derivative (7.2) and (7.3)

16 16 N.A. Gumerov & R. Duraiswami Fig Typica computationa mesh and output for the potentia on the sphere surface using the BEM (Comet 4., Coins & Aikman Automotive Interior Systems). become ψ S1 = Q 4πa 2 = ψ = iqσ 1 n 4πa S1 2 = (2 +1)h (kd) P (cos θ 1 ) kh (ka)+iσh, (7.5) (ka) (2 +1)h (kd) P (cos θ 1 ) kh (ka)+iσh (ka) = iσ ψ S1. (7.6) For the particuar case of a sound-hard (σ 1 =) surfaces this gives ψ S1 = Q 4πka 2 = (2 +1) h (kd) h (ka)p (cos θ 1 ), ψ =, (7.7) n Sq whie for sound-soft (σ 1 = ) surfaces we have ψ S1 =, ψ n S1 = Q 4πa 2 = (2 +1) h (kd) h (ka) P (cos θ 1 ). (7.8) These formuae are cassica and can be found esewhere (e.g. see Hanish, 1981, pp ) Resuts of Computations. To test our software MutisphereHemhotz for the case of a singe sphere we compared resuts of computations of the potentia on the surface with those provided by expression (7.7) and that obtained using the Boundary Eement Method (BEM), as reaized in the commercia software package COMET 4. from Coins and Aikman Automotive Interior Systems, Pymouth, MI. A typica output from this software package is shown in Fig The soution of the Hemhotz equation in a domain specified by a trianguar surface mesh, and prescribed ocations of the sources is shown. The figures beow iustrate comparisons of so-caed Head reated transfer function (HRTF) (see, e.g., Duda & Martens, 1998) computed using different methods. The HRTF is measured in db and represents the ratio of the ampitude of the acoustic fied at specified ocation of the surface to the ampitude of the incident fied at the center of the sphere: ψ HRTF =2g S1 ψ in (r 1 ). (7.9)

17 Mutipe Scattering from N Spheres 17 HRTF (db) σ/k = Frequency = 2 khz, Range = 1, Sphere Radius = 8.25 cm, ka = BEM, 1352 nodes, 27 eements Duda & Martens (1998), Axisymmetric MutiSphereHemhotz, Ntrunc= Incidence Ange, θ (degrees) Fig The head reated transfer function (HRTF) for a singe sphere of radius a =8.25 cm and a monopoe source, ocated at a distance d/a =1from the center of the sphere for spheres of different impedances σ/k (shown near curves). The continuous and dashed ines show resuts of computations using MutiSphereHemhotz with truncation number M =8. Circes and diamonds respectivey show the resuts of Duda & Martens (1998) and of BEM computations for σ =. In the atter the sphere surface was discretized using 27 inear eements. This function for a singe sphere was investigated by Duda & Martens (1998), who used soution (7.7) for this purpose. They aso provided a Matab code for computations of the HRTF with truncation based on evauation of the subsequent terms in the series (Duda, 2). Figure 7.2 shows good agreement between computations using a methods. In this exampe we incude aso computations for different impedances of the sphere. We aso tested the resuts of computations obtained using different truncation numbers. The origina code of Duda & Martens (1998) uses truncation based on comparison of subsequent terms in the series (in this particuar case the series was truncated when the ratio of such terms is smaer than 1 6 ). Experiments with MutisphereHemhotz showed that exceent agreement with these resuts is achieved, if the truncation number is seected using the formua (heuristic) M =[eka], e = (7.1) which shows that it increases with the wavenumber. For arge ka good agreement was observed for M > 1 2 eka,whie differences were visibe for smaer M. For moderate ka such differences appeared for truncation numbers beow the vaue provided in (7.1) Two Spheres. Since there is no cosed anaytica soution for two spheres in a simpe form, we compared numerica resuts obtained by using both our codes MutisphereHemhotz and Mutisphere- HemhotzCoaxia, and the BEM computations. As an exampe probem we considered computation of the HRTF for the so-caed Snowman mode, used for simpified study of the generation of head pose reated cues in an acoustic fied. The mode consists of two spheres, which are touching at one point. In computations the ratio of sphere

18 18 N.A. Gumerov & R. Duraiswami 1 5 Frequency = 2 khz Ntrunc=8 (auto) HRTF (db) Duda & Martens (1998) MutiSphereHemhotz MutisphereHemhotz, Ntrunc=4 Frequency = 2 khz Ntrunc=82 (auto) Incidence ange, θ (degrees) Fig Computations of the HRTF for a sound-hard sphere (σ =)usingtheduda&martens(1998) soution with evauation of the residua term in the series, and MutiSphereHemhotz with automatic seection of the truncation number and M =4(for frequency 2 khz). Other parameters are as in Fig 7.2. Fig An exampe of BEM (COMET 4.) computations of potentia distribution over the surface of two spheres (Snowman mode) generated by a monopoe source. Each sphere surface is discretized to 27 trianguar eements. The ratio of sphere radii is and they touch in one point. radii was taken to be The origin of the reference frame was ocated at the center of the smaer sphere ( the head ) and the direction of the y-axis was from the arger sphere ( torso ) to the head. The z axis was directed towards a monopoe source, generating the incident fied, which was ocated at the 2 Duda et a (21) have performed measurements with a simiar configuration. Comparison of simuations with these experiments wi be reported shorty.

19 Mutipe Scattering from N Spheres 19 HRTF (db) o 3 o θ 1 = o 9 o 6 o 18 o 12 o BEM MutisphereHemhotz 3 o 9 o 12 o -6 o o Two Spheres, ka 1 = Ange φ 1 (deg) Fig Comparison of the HRTF anguar dependence over the surface of a smaer sphere computed for the two sphere geometry shown in Fig. 7.4 using the BEM and Mutipoe reexpansion technique (MutipoeHemhotz). Both spheres have zero impedance. distance of 1 radii from the smaer sphere. The frequency of the incident wave nondimensionaized with the radius of the smaer sphere corresponded to ka 1 = The mesh for computations using BEM contained 54 trianguar eements (27 eements for each sphere) and was obtained by mapping of the surface of the cube onto each sphere. A picture of this Snowman with computationa mesh and distribution of the acoustic pressure is shown in Figure 7.4. In the computations the impedances of both spheres were set to zero ( sound-hard surfaces). For computations using MutisphereHemhotz the truncation number was automaticay set to 1 M = 2 ekr 12 (7.11) For given above vaues of a 2 /a 1 and ka 1 this provided M =9, so that the tota number of modes n in the mutipoe expansion was 1 for each sphere (or 1 A m n coefficients for each sphere). Figure 7.5 shows a comparison between the BEM and the MutisphereHemhotz computationa resuts for the HRTF cacuated for sphere 1 according equation (7.9). Each curve corresponds to a fixed vaue of the spherica poar ange θ 1 and demonstrates dependence on the ange ϕ 1. Note that dependence on the ange ϕ 1 is ony due to the presence of the sphere 2. Indeed in the absence of the torso, the potentia distribution over the sphere 1 surface is axisymmetric (see the previous case of a singe sphere), and there is no dependence on ϕ 1. The comparison shows a good agreement between the resuts obtained by different methods (the resuts produced by MutisphereHemhotz and MutisphereHemhotzCoaxia are on top of each other). Some sma dispersion of the points obtained using BEM is due to the mesh discretization of the sphere surface, which normay can be avoided by additiona smoothing/interpoation procedures. For the BEM resuts shown in the figure we did not appy such smoothing, but seected for potting the vaues of potentia at the eement centers ocated within some sma prescribed distance from a specified surface point (θ 1, ϕ 1 ). It is aso noticeabe that MutisphereHemhotz far outperformed BEM computationay, both in much higher speed and memory usage. Figure 7.6 demonstrate computations of the HRTF for the Snowman mode with the parameters described above, but for higher frequency, and different impedances of the arger sphere (which can somehow mode

20 2 N.A. Gumerov & R. Duraiswami 8 Two spheres ka 1 =1, σ 1 =, θ 1 =6 o 6 HRTF (db) 4 2 σ 2 /k: Ange, φ 1 (deg) Fig Anguar dependence of the HRTF over the surface of the sound-hard smaer sphere for the geometry shown in Fig. 7.4 for different impedances of the arger sphere. Resuts computed using the MutisphereHemhotz code with the automaticay seected truncation number N t =31. the presence of cothes). For the given geometry parameters the automaticay seected truncation number was N t =31. This number is arge enough to observe a substantia difference in speed of computations and memory usage by MutisphereHemhotz and MutisphereHemhotzCoaxia (see evauation (5.14), where M = N t ). Proper seection of the truncation number is important issue for appications of mutipoe transation techniques. Figure 7.7 shows convergence of computations with increasing truncation numbers for the HRTF at any specified point on the surface (θ 1 =6 and ϕ 1 = inthecaseshowninthefigure). Computations with ow truncation numbers may provide poor accuracy. At some truncation number (which depends on the frequency) the computationa resuts stabiize (note that the HRTF depends on the frequency, and so for each frequency we have its own horizonta asymptote). Further increase of the truncation number increases both the accuracy of the resuts and computationa time/memory, since the matrix size grows proportionay as Nt 4 in the genera case, and as Nt 3 for coaxia spheres (so there is a trade-off issue). However, at some truncation numbers, which sighty exceed the vaue provided by (7.11), computations usuay encounter difficuties connected with exponentia growth of portions of the terms in the expansions, eading to overfow reated errors. These are the spherica Hanke functions of arge order, h n kr pq, entering the reexpansion system matrix. Asymptotic expansion of the Hanke function at arge n and fixed kr12 shows, that the growth starts at n 1 2 ekr 12. This is used as the basis for automatic seection of the truncation number (7.11). Of course, this imitation is purey based on the order of computation, and the product of terms remain finite, and cacuations can be performed for arger N t than given by (7.11). To deveop software using such sums of finite terms additiona study of the truncated tais of the series at arge n is needed, and a proper impementation remains to be deveoped. The computations presented in Figure 7.7 show that the actua stabiization occurs at smaer N t sthan theonegivenby(7.11)(wherewehaveforka 1 =1, 5, 1, 2, 3 the foowing vaues: N t =3, 15, 31, 63, 94, respectivey. Resuts of our numerica experiments show that for arge kr12 reasonabe accuracy can be achieved at M 1 2 M max, where M max is provided by equation (7.11). At the same time, for ower kr12 formua (7.11) provides vaues which cannot be reduced, and accurate computations can be achieved with M sighty arger than M min, with M min is provided by equation (7.11).

21 Mutipe Scattering from N Spheres HRTF (db) -1-2 ka 1 =3 Two spheres, θ 1 = 6 o, φ 1 = o, r min /a 1 = Truncation Number Fig Dependences of compuations of the HRTF for two sphere geometry (see Fig. 7.4) on the mutipoe series truncation number for different non-dimensiona wavenumbers. Note that in our previous paper (GD21b) we aso presented resuts of the computation of the HRTF for a sphere near a rigid wa. In this case the rigid wa coud be repaced by an image sphere and an image source and the coaxia mutipoe reexpansion can be used. The probem of sound scattering by a sphere near a rigid wa is in some sense a simpified genera probem for two spheres, since both the rea and the image spheres in this case have the same radius and impedance. For this case, the coefficients of the mutipoe expansions near each sphere A (1)m n and A (2)m n are symmetrica and the dimension of the system can be reduced using this symmetry by a factor of two. In that paper, we aso provided a study and discussion of the infuence of the distance between the sphere and the wa and frequency of the fiedonthehrtf. Seected resuts of the HRTF for the Snowman mode incuding comparisons with experimenta data were aso presented in Gumerov et a (21) Three Arbitrariy Located Spheres. If the cases of one and two spheres can be covered using simpified codes, the case of three non-coaxia spheres requires the genera three-dimensiona mutipoe transation deveoped in GD21. As in the case of two spheres discussed above, we compared resuts of computations for three spheres using MutisphereHemhotz and the COMET commercia BEM software. For this computationa exampe we paced an additiona sphere (#3) to the case of the Snowman mode, described above. The radius of the sphere was equa to the radius of the head, and it may mode some object of the size of the head paced between the sound source and the istener. The distance between the centers of spheres #1 and #3 was the same as the distance between the centers of spheres #1 and #2. The parameters of the incident fied were the same as.f orthe snowman case. The mesh for computations using the BEM contained 5184 trianguar eements, 1728 eements for each sphere and was obtained by mapping of the surface of the cube to each sphere. A picture of this configuration with computationa mesh and distribution of the acoustic pressure is shown in Figure 7.8. In computations the impedances of a three spheres were set to zero. Resuts of comparisons between BEM and MutisphereHemhotz computations with M =9are shown in Figure 7.9. The comparison is as good as in the case of two spheres. Since Figures 7.5 and 7.9 represent simiar dependences, we can notice that the presence of the third sphere reduced (at some points by 3-4 db) the ampitude of the sound fied on sphere 1. This is a cear physica effect, since sphere 1 was situated in the acoustic shadow of sphere 3.

22 22 N.A. Gumerov & R. Duraiswami Fig An exampe of BEM (COMET 4.) computations of the potentia distribution over the surface of three spheres generated by a monopoe source. Each sphere surface is discretized to 1728 trianguar eements. Two spheres of reative radii 1 (#1) and (#2) touch in one point and the ceneter of the third sphere (#3) of reative radius 1 is ocated at the distance from the center of sphere #1 ontheineconnectedthesourceandthecenterofsphere#1. HRTF (db) o 3 o θ 1 = o 9 o 6 18 o BEM MutisphereHemhotz 3 o o 9 o 12 o o 15 o o Three Spheres, ka 1 = Ange φ 1 (deg) Fig Comparison of the HRTF anguar dependence over the surface of smaer sphere computed for the three sphere geometry shown in Fig. 7.4 using the BEM and Mutipoe reexpansion technique (MutipoeHemhotz). A three spheres have zero impedance.