Effects of surface scattering and room shape on the correspondence between statistical- and geometrical-acoustics model predictions.

Abstract

Much of room acoustics relies on approximate models of sound. The high-frequency approximation of geometrical acoustics, in which sound is modeled by ensembles of classical phonons, is the basis for many computational modeling techniques. The earliest mathematical model of sound in rooms, given by Sabine's decay equation, is a statistical approximation that assumes homogenous and isotropic (diffuse) sound fields and continuous absorption processes. Sabine's statistical-acoustics model is related to geometrical acoustics as a limiting case for rooms that are ergodic, sufficiently mixing, and weakly absorbing. Various semi-empirical corrections are possible (e.g., Eyring), though a true first-order correction requires additional information. Surface scattering (as characterized by scattering and diffusion coefficients) and room shape are critical in determining the relationship between statistical- and geometrical-acoustics predictions, as these physical attributes determine key statistical properties of phonon trajectories. Here, this relationship is examined theoretically. Limiting conditions are described under which various statistical-acoustics models are valid. Underlying assumptions of Sabine's and Eyring's decay models are detailed. For more general conditions, a series of heuristics are outlined. Based on this discussion, it is shown that statistical acoustics cannot yield an internally consistent formula for relating surface scattering to sound-field diffusivity.