Synergies between the high-frequency Boundary Element Method and Geometric Acoustics

Overview & Synopsis

The audible frequency range covers many octaves in which wavelength changes from being large with respect to dominant features of a space to being comparatively much smaller. This makes numerical prediction of a space?s acoustic response, e.g. for auralisation, extremely challenging if all frequencies are to be represented accurately. Different classes of algorithm give the best balance of accuracy to computational cost in different frequency bands. At low frequencies, wave effects such as diffraction and interference are essential, but methods modelling the underlying PDEs directly have computational cost that scales with problem size and frequency, rendering them inefficient or intractable at high frequencies. At high frequencies, Geometrical Acoustics descriptions are more efficient, but the accuracy they can achieve is limited by how well the geometric ray assumption represents sound propagation in a given scenario; this compromises accuracy at low frequencies in particular. It is therefore often necessary to operate two algorithms in parallel handling different bandwidths. Combining their output data can however be an awkward process due to their differing formulations. This is particularly important for early reflections, which give crucial spatial perceptual cues ? for late time the wave field becomes chaotic at high frequencies and the benefits are less clear. There is therefore a need for a unified full audible bandwidth algorithm for early reflections. This paper will describe ongoing research to develop such an algorithm by exploiting synergies between Boundary Element Method (BEM) and Geometric Acoustics (GA). It will describe how appropriately chosen oscillatory basis functions in BEM can produce leading-order GA behaviour at high frequencies, and how these might be assembled into a full solution. It will introduce the Wave Matching BEM, a new formulation that can be solved by marching-on-in-reflection order, a property shared with GA that makes it inherently suitable for modelling early reflections.