Including scattering within the room acoustics diffusion model: An analytical approach

Abstract

Over the last 20 years, a statistical acoustic model has been developed to predict the reverberant sound field in buildings. This model is based on the assumption that the propagation of the reverberant sound field follows a transport process and, as an approximation, a diffusion process that can be easily solved numerically. This model, initially designed and validated for rooms with purely diffuse reflections, is extended in the present study to mixed reflections, with a proportion of specular and diffuse reflections defined by a scattering coefficient. The proposed mathematical developments lead to an analytical expression of the diffusion constant that is a function of the scattering coefficient, but also on the absorption coefficient of the walls. The results obtained with this extended diffusion model are then compared with the classical diffusion model, as well as with a sound particles tracing approach considering mixed wall reflections. The comparison shows a good agreement for long rooms with uniform low absorption (α = 0.01) and uniform scattering. For a larger absorption (α = 0.1), the agreement is moderate, due to the fact that the proposed expression of the diffusion coefficient does not vary spatially. In addition, the proposed model is for now limited to uniform diffusion and should be extended in the future to more general cases.

An Improved Acoustic Diffusion Equation Model for Long-Channel Underground Spaces

“…This idea was further applied to coupled spaces by optimizing the boundary conditions [ 32 ]. Foy et al [ 33 ] extended the ADE model by modifying the absorption and scattering coefficients. This extension improved the prediction accuracy, but the more complicated expression increased the computational load, and its application is limited.…”

3-D Diffusion Models for Predicting Reverberant Electromagnetic Power Density in Loaded Enclosures

“…where Aw and Ac are the surface areas of the walls and contents and V is the volume of the enclosure. The overall mean free path is given by: (8) On the boundary surface of the enclosure walls and contents, the energy density is assumed to satisfy a Robin boundary condition: (9) where is the outward unit normal vector and is the absorption factor of the walls. The simplest estimation of is obtained by Sabine's formula [16]: (10) where w( ) is the absorption efficiency of the walls.…”

“…We use the FreeFEM++ software to build the 3D diffusion model [22]. The walls and contents of the enclosure are modeled by including their surfaces in the mesh and applying the Robin boundary condition (9), with the appropriate absorption efficiency. Fig.…”

“…The latest development of the diffusion model can be found in [9] and [10]. The diffusion method stems from the radiative transport theory of rays in an enclosure and can be seen as a generalization of the PWB method.…”

A mixed boundary condition for accounting energy absorption and transmission in a diffusion equation model

“…In Ref. 3, the acoustic radiative transfer method is used to extend the diffusion model to be applied with absorption boundary conditions, combining a proportion of specular and diffuse reflections. Moreover, another energetic wave equation model in room acoustics has been proposed from the propagation equation of radiative transfer.…”