Proc. of the 2nd International Symposium on Ambisonics and Spherical Acoustics May 6-7, 2010, Paris, France A COMPARATIVE STUDY OF PLATONIC SOLID LOUDSPEAKERS AS DIRECTIVITY CONTROLLED SOUND SOURCES Alexander Mattioli Pasqual Philippe Herzog Jose´ Roberto de Franc¸a Arruda Faculty of Mechanical Engineering Laboratory of Mechanics and Acoustics State University of Campinas National Center for Scientific Research Rua Mendeleiev 200 31 chemin Joseph-Aiguier 13083-970, Campinas – SP, Brazil 13402 Marseille cedex 20 France [pasqual,arruda]@fem.unicamp.br [email protected] ABSTRACT ARMs are real orthogonal functions (or vectors, if the vibrating body possesses a finite number of degrees of freedom) describ- Compact spherical loudspeaker arrays in the shape of Platonic ing velocity patterns over the surface of a vibrating body. As far solids have been used as directivity controlled sound sources. as compact loudspeaker arrays are concerned, one of the main Such devices are commonly used to synthesize spherical har- advantages of the ARMs over the SHs is that they span a fi- monic (SH) radiation patterns, which might be later combined nite dimension subspace (the subspace dimension is the number to produce different directivities. The SH synthesis presents of drivers in the array) on which any radiation pattern the loud- two main difficulties whose extent depends on the Platonic solid speaker array is able to produce can be projected with no approx- used: large synthesis error at high frequencies due to spatial imation error, whereas SHs span an infinite dimension subspace aliasing and huge membrane velocities at low frequencies due so that truncation error generally arises. On the other hand, un- to low radiation efficiency. This work presents a comparative like the ARMs, the SHs lead to a directivity format that does not theoretical study of the five Platonic loudspeakers. Their ability depend on the reproduction device and therefore the sound mate- to reproduce SHs and the radiation efficiencies of their acoustic rial so encoded is compatible to different loudspeaker arrays [8]. radiation modes are evaluated. It is shown that the dodecahe- In fact, the real SHs are the ARMs of the continuous dron presents the best compromise between complexity of the sphere [9], which is defined here as the sphere able to assume controllable patterns, number of channels and sound power. any surface velocity pattern (infinite number of degrees of free- dom). Moreover, at low frequencies, the ARMs of spherical ar- 1. INTRODUCTION rays give rise to far-field radiation patterns that match the real SHs, although the radiation efficiencies of an ARM and its cor- Compact spherical arrays of independent loudspeakers operating responding SH are different [7]. In other words, at low frequen- at the same frequency range have been used as directivity con- cies, the synthesis error in mimicking a given target directivity in trolled sound sources. The transducers are usually distributed the far-field remains unchanged whether the ARMs or the SHs over the spherical frame according to a Platonic solid geometry are used. However, only the ARMs provide the correct radiation to obtain a highly symmetrical configuration, so that the occur- efficiencies associated to each channel of the loudspeaker array. rence of preferred regions in the 3-D rendition space is reduced. This work presents a comparative theoretical study of the Hexahedral [1], dodecahedral [2, 3] and icosahedral [4] five Platonic solid loudspeakers (with one driver per face) as di- sources with one driver per face have been reported in the liter- rectivity controlled sound sources. Their radiation patterns are ature (an icosahedron with six drivers per face was proposed in derived analytically by approximating each driver as a convex [5]). Such devices are commonly used to synthesize pure spher- spherical cap that oscillates with a constant radial velocity am- ical harmonics (SHs) in the far-field under free-field conditions, plitude over its surface, as described in [6]. The ability of the which might be later combined to produce different directivities. Platonic sources in reproducing pure SH patterns in the least- The synthesis of pure SHs by a compact loudspeaker array squares sense is reviewed. In addition, the radiation efficiencies presents two main difficulties, namely, spatial aliasing in the of the ARMs of each Platonic source are computed in order to high-frequency range and low radiation efficiency in the low- evaluate the low-frequency constraint mentioned before. frequency range. The former degrades the SHs synthesis [1, 2,6] and the latter implies huge membrane velocities in order to pro- duce meaningful sound power levels [7]. The extent of these 2. THEORY problems is greatly affected by the position of each driver on the array and by the number of transducers, i.e., the choice of the This section gives an overview of the main theory related to the Platonic solid is of major importance to the array performance. radiation control by a compact spherical loudspeaker array. Instead of using SHs as preprogrammed basic directivities, the acoustic radiation modes (ARMs) of the loudspeaker array 2.1. Acoustic radiation modes (ARMs) can be used as a basis for directivity representation [7]. The This subsection presents a short introduction to the ARMs. For This work was sponsored by CAPES (Brazil) and CNRS (France). further details, refer to [7, 9]. Proc. of the 2nd International Symposium on Ambisonics and Spherical Acoustics May 6-7, 2010, Paris, France For a vibrating body with L degrees of freedom, the ARMs where mn = n2 + n +1+ m is used for linear indexing of the (1,2) form a set of L real orthogonal vectors that span the subspace SHs, k is the wave number, a is the sphere radius and hn ( ) of the achievable velocity patterns over the body surface. The are the spherical Hankel functions of the first and second kind.· ARMs can be obtained by the eigenvalue decomposition of an Equation (2) shows that σmn does not depend on m and is operator related to the radiation efficiency of the vibrating body. only a function of the non-dimensional parameter ka for SHs of The radiation efficiency, σ, of a vibrating structure with L a given order n. The radiation efficiencies for the first 49 ARMs degrees of freedom can be written as (SHs up to order n = 6) are presented in Fig. 1. H u Cu 0 σ(u)= (1) 10 uHVu where u is a column vector of velocity amplitude coefficients, −5 C is an L L real symmetric matrix which couples the power 10 radiated by× the elements of u, V is an L L real positive diag- × onal matrix which leads to the spatial mean-square velocity on mn −10 10 the structure surface and the superscript H indicates the complex σ mn=1 (n=0) mn=2−4 (n=1) conjugate transpose. Throughout this work, unless otherwise mn=5−9 (n=2) −15 specified, lower case bold letters indicate vectors, while upper 10 mn=10−16 (n=3) case bold letters indicate matrices. mn=17−25 (n=4) mn=26−36 (n=5) Notice that the radiation efficiency is in the form of the gen- mn=37−49 (n=6) −20 eralized Rayleigh quotient. Thus, the solution of the generalized 10 −2 −1 0 1 eigenvalue problem Cψ Vψ leads to a set of real orthog- 10 10 10 10 = λ L ka onal eigenvectors ψ1, ψ2,..., ψL corresponding to real eigen- values, ordered as λ1 λ2 . λL. These eigenvectors ≥ ≥ ≥ Figure 1: Radiation efficiencies of the first 49 acoustic radiation are the ARMs and the eigenvalues are their radiation efficiency modes of the continuous sphere (spherical harmonics). coefficients, i.e., σl σ(ψ )= λl. ≡ l It can be shown that these modes (ARMs) radiate sound en- Figure 1 illustrates the grouping characteristic of the acous- ergy independently, i.e., the total radiated sound power is given tic radiation modes discussed in [9]. The number of modes by a linear combination of the sound power produced by each within each group is 2n+1, i.e.,1(n = 0),3(n = 1),5(n = 2), mode. Because such a modal decomposition is closely related 7(n = 3), 9(n = 4), etc. It is shown that σmn increases with to the radiation efficiency of the vibrating body, it permits rank- ka and decreases with n. Moreover, at low ka values, the radia- ing the expansion terms (ARMs) with respect to their radiation tion efficiency is strongly affected by n, so that simple directiv- efficiencies, so that a reduced number of active channels can be ity patterns (lower order SHs) radiate much more efficiently than obtained by not driving inefficient modes. In addition, the ARMs complex ones. This result pinpoints the main difficulty concern- are only a function of frequency, the radiating structure geome- ing sound radiation reproduction in the low-frequency range: the try and the constraints the body is subjected to; they do not de- sphere surface must present a huge velocity amplitude in order pend on the source of excitation and on the mass and stiffness of to produce complex directivity patterns with meaningful sound the structure. Furthermore, the ARMs of some radiators (e.g., a power levels. continuous sphere) are proven to be frequency independent. For It is worth noting that σmn can be increased over a given these reasons, the ARMs are a useful representation of vibration frequency range by increasing a. However, for compact spheri- patterns when one is mainly interested in the sound field radiated cal loudspeaker arrays, it is known that a larger sphere limits the by a vibrating structure. radiation control in the high-frequency range due to spatial alias- ing artifacts.