Prediction of the Reverberation Time in Rooms with Non Uniform Absorptions Using a Diffusion Model

The acoustical comfort is now a comfort’s criterion which is taken into from the conception. The designers need therefore accurate tools to predict the acoustic quality of enclosures. Most of the acoustical comfort criteria are based on the reverberation time. When the room has proportionate dimensions and an uniform absorption, the statistical theory through Sabine or Eyring formulas allows good predictions of the reverberation time. Moreover, extensions of these relations by Millington or Cremer and Müller among others give quite satisfactory when the room is composed of materials with different sound absorption. In this study, the reverberation time in an enclosure is calculated via the numerical resolution of unstationary diffusion equation, model validated in coupled and industrial rooms. Firstly, an improvement of the boundary condition is proposed for highly absorbent surfaces. The diffusion model is then compared to several formulations of the statistical theory and a raytracing software for a cubic room with homogeneous walls’ absorption and with non homogeneous walls’ absorptions. Finally, an experimental validation is conducted for an enclosure with non uniformly distributed absorption. INTRODUCTION Reverberation time is one the key criteria to qualify the sound quality of an enclosure. Its prediction before the building construction is then needed. Sabine [1] and Eyring [2] proposed very simple relations between the acoustical and geometrical outlines of an enclosure and its reverberation time. These relations give birth to the statistical theory and are the most accurate for rooms with quasi-cubic shape and homogeneous absorption and, for Sabine’s one, limited to low absorptions. However, in most A. Billon and A. Sakout applications, the sound absorption of walls is inhomogeneous and the shape differs greatly from a cube. The Sabine’s and Eyring’s models have been refined over the years and other methods have been developed like ray, beam, cone and sound particle tracing, integral equation method, image-source method and for low frequencies, modal theory. Recently, a model has been proposed in which a diffusion equation is solved numerically to obtain the reverberant sound field [3]. In this study, this model is modified and compared to several models derived from the statistical theory, the diffusion equation based model, and a commercial ray-tracing software, CATTAcoustic, to experiments for a quasi-cubic room fitted with non-uniformly distributed absorptions. Firstly, several models based on Eyring and Sabine formulations used in this paper are presented. Then, the diffusion model is introduced and a modification is proposed to improve its predictions for the high absorptions. To validate the modification of the diffusion model, numerical comparisons are carried out in Section 3 with statistical theory based models and the ray tracing software in a cube with homogeneous absorptions. In Section 4, an experimental set-up is presented and its results are compared to the models predictions. MODELS PRESENTATION In this section, several models based on the statistical theory are presented. In a second time, a recently proposed model based on the numerical solving of a diffusion equation is showed. A modification of the boundary condition for high absorption coefficient is then proposed. Statistical theory based models For rooms with homogeneous dimensions and absorption, Sabine [1] obtained an expression of the reverberation time,