A Geometrical Acoustics Approach Linking Surface Scattering and Reverberation in Room Acoustics

A general model of the influence of surface scattering on the reverberation time would have several applications in room acoustics. Such a model is not yet available, and it is the purpose of this paper to investigate a geometrical acoustics approach. Starting from the radiative transfer equation, an exponential solution is first developed for the reverberation energy decay in rooms with diffusely reflecting boundaries. Differences with the diffuse sound field (Sabine) theory are highlighted, leading to a modified formula for the reverberation time which is shown to be more in accordance with ray tracing simulations. A general model is then proposed for rooms in which specular and diffuse reflections coexist. This general model is applied in rooms where the specular contribution can be assumed to be quasi-isotropic and uniform. Under this assumption, the reverberation decay is represented by the sum of two exponential functions, depending on the scattering coefficients. However, it is shown that the influence of surfaces' scattering on reverberation is rather limited in this case. On the contrary, rooms with a pair of parallel surfaces are prone to create significant anisotropy in the cloud of image sources. An analytical formulation is proposed in this case for the specular and diffuse contributions, provided that some assumptions are again made on the specularly reflected sound field. The final expression is not really intuitive concerning the relation between scattering coefficients and reverberation, but it contains all the variables influencing this relation. This allows fast evaluations of the effect of surface scattering in particular situations. Finally, the application of this model to room acoustics computer simulations is illustrated by an example.