Numerical geometric acoustics

Abstract

Sound propagation in air is accurately described by a small perturbation of the ambient pressure away from a quiescent state. This is the realm of linear acoustics, where the propagation of a time-harmonic wave can be modeled using the Helmholtz equation. When the wavelength is small relative to the size of a scattering obstacle, techniques from geometric optics are applicable. Geometric methods such as raytracing are often used for computational room acoustics simulations in situations where the geometry of the built environment is sufficiently complicated. At the same time, the high-frequency approximation of the Helmholtz equation is described by two partial differential equations: the eikonal equation, whose solution gives the first arrival time of a geometric acoustics/optics wavefront as a field; and a transport equation, the solution of which describes the amplitude of that wavefield. Phenomena related to high-frequency acoustic diffraction are frequently omitted from these models because of their complexity. These phenomena can be modeled using a high-frequency diffraction theory, such as the uniform theory of diffraction. Despite their shortcomings, geometric methods for room acoustics provide a useful trade-off between realism and computational efficiency.