Energetic wave equation for diffuse sound fields

Abstract

The diffusion equation for modeling diffuse sound fields was proposed some fifty years ago on heuristic principles as an extension to Sabine’s diffuse field model, and still receives much attention. In a recent publication (Acta Acustica 103 (2017) 480–491), the authors developed the model one step further, using the full stress-energy tensor to provide the missing relations between sound intensity and sound energy. This introduces some extra terms that, in case of non-Sabine spaces (narrow or flat rooms), can be defined with the help of the boundary conditions in terms of absorption and scattering coefficients on the walls. Integrating the divergence of the stress-energy tensor across the shortest dimensions of the space leads to a propagation equation of the telegrapher type, which can be solved using finite difference time domain simulation. Schemes for one-dimensional (corridors) and two-dimensional (open-space) spaces are proposed, and numerical results compared to measurements in real spaces. The comparison makes it possible to evaluate the absorption and scattering coefficients by an adjustment procedure. The paper discusses the range of values taken by these coefficients and compares them to more traditional building-acoustical coefficients. It also presents under which further assumptions the diffusion equation is recovered.