INTERNATIONAL CONGRESS ON ACOUSTICS MADRID , 2-7 SEPTEMBER 2007 DIFFUSION-BASED MODELS FOR PREDICTING SOUND FIELDS IN ROOMS WITH MIXED SPECULAR AND DIFFUSE REFLECTIONS PACS : 43

A diffusion model, based on the numerical solution of a diffusion equation, was developed in the past to give accurate predictions of the sound-pressure level and of the sound decay at any location in a single enclosure or in coupled rooms. However, this model is limited to rooms with diffusely reflecting walls. In this study, two methods are presented to extend this model to rooms with mixed specular and diffuse reflections, defined by the classical room-acoustics scattering coefficient. The first method models the reverberant field by using a modified diffusion process, with an empirical diffusion constant departing from the theoretical one based on the mean free path. The set of these coefficients, which depend on the scattering coefficient, has been found empirically to fit the results given by a ray-tracing program. The second method is a hybrid approach. The reverberant sound field due to the diffuse reflections is first obtained with the original diffusion model. Then the sound field due to the specular reflections is modelled with an image-source model. The results given by the two approaches are compared to experimental data obtained for several types of rooms (a classroom, an office and some long rooms). INTRODUCTION Over the last years, a new model, based on a diffusion process, has been developed to predict the reverberant acoustic field in rooms [1,2]: a well-identified diffusion process, taking into account the room geometry (via the so-called diffusion constant), the surface absorption (via the boundary conditions) and the source power and location, is solved numerically to obtain accurate predictions of the sound-pressure level and of the temporal sound decay at any location in a single enclosure (empty or fitted with scattering objects [3]) or in coupled rooms [4]. The results obtained have been shown to be in good agreement with results given by the radiosity method [1,5], ray-tracing models [2,3,4] and experimental data [3,4]. A main advantage of this numerical method is that it can be applied to spaces with complex shapes (provided that a mean free path can be estimated), such as coupled spaces, and that the calculation times involved are very low. However, this so-called “diffusion model” is intrinsically limited to rooms with diffusely-reflecting boundaries (i.e., the reflection law is described by Lambert’s law [5]), and therefore has been validated only for diffusely-reflecting rooms. The aim of this study is to show that this model can be adapted to take into account mixed types of reflection – i.e., with a given proportion of diffuse and specular reflections. This issue is indeed of critical importance, as it has been shown that sound fields in “real rooms” are the result of such mixed reflections [6]. Two alternative versions of the diffusion model are proposed. The first, called in the following the “empirical diffusion model”, is based on a modified diffusion process, in which the diffusion constant has been adjusted empirically to model any given proportion of specular reflections. The second is a “hybrid model”: the reverberant field due to the diffuse reflections is calculated by using the original diffusion model, and the one due to the specular reflections is calculated by using the image-source model [5]. These two models are presented in the next section. Following that, the numerical results are compared with experimental data, in terms of spatial variations of sound pressure level (SPL) and reverberation time (RT). The cases considered are two rooms with homogeneous dimensions, and an elongated room. In the last section, conclusions are drawn about the performance of the proposed models. DIFFUSION-BASED MODELS The theoretical diffusion model [1,2] Let us consider a single empty volume in which perfectly diffuse sound reflections are assumed. The input parameters for the room-acoustic diffusion model are the following: • V and S, the room volume and the area of its surfaces, respectively, and consequently its mean free path, λ = 4V/S; • the complete geometry of the room surfaces; • the local absorption coefficients of the surfaces α. Arbitrary spatial variations of this coefficient are possible; • an arbitrary number N of omnidirectional sound sources assumed to be points, with output acoustic power Pi(t) and location ri (i = 1.N); • the frequency-dependent atmospheric attenuation coefficient m (in m). The timeand space-dependent reverberant acoustic energy density, w(r,t), is then governed by the following diffusion equation : ). ( ) ( ) , ( ) , ( ) , ( 1 ∑ = − = + Δ − ∂ ∂ N i i i r r t P t r mcw t r w D t t r w δ (Eq. 1) In this equation, Δ is the Laplace operator. D is the diffusion coefficient, with D= λc/3 (c is the speed of sound); it takes into account the room geometry, via the mean free path λ, and is theoretically valid only for diffuse reflections. The diffusion equation of Eq. 1 is solved numerically (by using the finite-element method), together with the following boundary conditions: ( ) ln 1 ( , ) ( , ) 0 4 c w r t D t α − ∂ − ∂ w r t = (Eq. 2) This equation models the sound absorption by the room surfaces. The solution w(r,t) of the system of Eqs. (1) and (2) permits the RT and SPL to be predicted at any location in the room [2]. The empirical diffusion model A modified diffusion model has been derived to permit the modelling of rooms with mixed reflections. In the following, these mixed reflections are characterised by the well-known scattering coefficient s, defining the proportion of energy reflected in a diffuse manner. This coefficient ranges between 0 (for totally specular reflections) and 1 (for totally diffuse reflections). The development of this modified diffusion model follows from numerical observations: the spatial variations of the SPL in a given room, as obtained using a ray-tracing model for several values of the scattering coefficient s, can be modelled by a diffusion equation, provided that the diffusion constant is properly “empirically” adjusted [3] to a value Demp such that: D K Demp = , (Eq. 3) D being the theoretical value of the diffusion constant (λc/3), valid for purely diffuse reflections. The problem is how to find K as a function of s. 19 INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 2 An example of these observations is shown in Figure 1. The sound field is calculated using a ray-tracing model for a long room of length 15m and cross-section (2×3) m, and for values of s of 0.2, 0.6 and 1.0. Results for the variation of SPL with position are plotted in Figure 1a, along the longer dimension of the room (from x=0 to x=15m; the source with a power level of 100 dB is located at x=1m). Diffusion-model results are shown in Figure 1b: the value of the adjustment constant K (defined in Eq. 3) has been adjusted so that the SPL curves (calculated from the solution of the diffusion equation) match those given by ray-tracing; the corresponding K values are 5.72, 2.54 and 1.00, respectively. The SPL curves given by the two methods behave in a very similar manner [6]: the curves cross each other at about x=5m. Further from the source, the SPL increases if the proportion of specular reflections increases – i.e., if s decreases or, equivalently, Demp=KD increases. These observations show that the SPL in a room with mixed reflections can be modelled by a well-identified diffusion process. The adjustment of the diffusion constant has been carried out for a set of rooms covering a wide range of aspect ratios. It was shown that the adjustment constant K is not a function of the aspect ratio of the room, but only of the scattering coefficient s. 0 5 10 15 86 88 90 92 94 96 98 100