An Energy Interpretation of the Kirchhoff-Helmholtz Boundary Integral Equation and its Application to Sound Field Synthesis

Abstract

Most spatial audio reproduction systems have the constraint that all loudspeakers must be equidistant from the listener, a property which is difficult to achieve in real rooms. In traditional Ambisonics this arises because the spherical harmonic functions, which are used to encode the spatial sound-field, are orthonormal over a sphere and because loudspeaker proximity is not fully addressed. Recently, significant progress to lift this restriction has been made through the theory of sound field synthesis, which formalizes various spatial audio systems in a mathematical framework based on the single layer potential. This approach has shown many benefits but the theory, which treats audio rendering as a sound-soft scattering problem, can appear one step removed from the physical reality and also possesses frequencies where the solution is non-unique. In the time-domain Boundary Element Method approaches to address such non-uniqueness amount to statements which test the flow of acoustic energy rather than considering pressure alone. This paper applies that notion to spatial audio rendering by re-examining the Kirchhoff-Helmholtz integral equation as a wave-matching metric, and suggests a physical interpretation of its kernel in terms of common acoustic power flux density between waves. It is shown that the spherical basis functions (spherical harmonics multiplied by spherical Bessel or Hankel functions) are orthogonal over any arbitrary surface with respect to this metric. Finally other applications are discussed, including design of high-order microphone arrays and the coupling of virtual acoustic models to auralization hardware.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…Here the combination of inner product terms used in the testing integral was shown to have a 'Wave-Matching' property; given a trial wave (the testing function), it will return a coefficient proportional to the amplitude of that wave component within whatever field it receives. This form was also identified for use in microphone arrays in [54] and has previously been presented by the authors of this paper in three dimensions in [57]. This led to a set of orthogonality relations between the incoming and outgoing waves, which makes the interaction matrices diagonal.…”

“…This is done in [54] to eradicate 'front-back confusion' in planar arrays, and in [54][55][56] with spherical arrays to distinguish between waves arriving from outside or emanating from inside the array. Green's theorem is used to extend this to arbitrary shaped arrays in [57]; a similar formulation to this will be derived in two dimensions herein.…”

“…This orthogonality applies both in order , which also represents surface-tangential wavenumber, and whether the waves are incoming or outgoing. An equivalent property was demonstrated for spherical waves in 3D in [57]. Substitution of gives:…”

“…The algorithm would ideally also be taken into three-dimensions. Some parts of the framework for this are already in place [16,57], but many challenges remain, hence this paper has restricted itself to twodimensional problems. Efficient integration is a challenge that restricts the effectiveness of all these applications, but much work is being done to address that.…”

The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

“…For some papers, it was impossible to decide whether the authors chose the better solution or not. 8,35,36,38,57 In these cases, some information for the decision has been missing in the paper. Although being the linear combination of two zeros, cf.…”