A numerical investigation of the Fick's law of diffusion in room acoustics

Abstract

In this paper the validity of the Fick's law of diffusion in room acoustics is investigated in the stationary state. The Fick's law, underlying the room-acoustics diffusion model, assumes a proportionality relationship between the local sound intensity and the energy density gradient, the proportionality constant being the so-called diffusion coefficient. This relationship, based on an analogy with the behavior of real particles in a scattering medium, is assessed by using a numerical tool simulating the actual dynamics of sound particles in a room. Two types of room geometries are considered: rooms with proportionate dimensions and long rooms. Concerning proportionate rooms the numerical analysis highlights the presence of weak variations of the reverberant energy density, generating an intensity vector pattern which is shown to be correctly described by the theoretical Fick's law and homogeneous diffusion. Conversely, inside long rooms, an estimate of the local value of the diffusion coefficient is carried out, showing that the reverberant sound field is well described by a spatially varying diffusion coefficient (non-homogeneous diffusion). The rate of increase of the estimated diffusion coefficient depends on the cross-sectional area of the room and on the boundaries absorption coefficient.