The acoustic diffusion model is based on the assumption that the volumes (rooms) studied contain scatterers that uniformly scatter the sound field and that the sound field is diffuse (large number of reflections). Using the diffusion of light in a scattering environment as an analogy one can express a diffusion equation for the sound-energy density w = w(x,t) (SI unit: J/m3). The diffusion equation describes the energy flow from high to low energy regions. Further details about the model equations and boundary conditions are found in papers by Xiang and others (see Ref. 3, Ref. 7, and Ref. 8) and in papers by Billon, Valeau, and more (see Ref. 5, Ref. 6, Ref. 11, and Ref. 15).

The domain diffusion equation for the sound-energy density w = w(x,t) is given by

where the local energy flux vector J (SI unit: J/m2/s = W/m2) is defined in the usual way, as

The total diffusion coefficient is Dt = D = λc/3 (SI unit: m2/s), λ is the mean free path (SI unit: m), c is the speed of sound (SI unit: m/s), and ma is the volumetric absorption coefficient of air (SI unit: 1/m). The volumetric absorption coefficient (or attenuation coefficient) should not be confused with the energy absorption coefficient α used in boundary conditions which is dimensionless. The source term q represents the spatial sound source (SI unit: J/m3/s = W/m3). The term cmaw accounts for volume absorption in air (dissipation). Note that in certain models, the cmaw term accounts for the total absorption at boundaries in a “mean” sense (only using no flux boundary conditions); this approach is not used here.

In the interface, ma can be given as a user input or it can be defined as the “classical viscous and thermal (volumetric) absorption” coefficient (as in pressure acoustics) given by

The absorption coefficient is either integrated over the band or it is given at a certain frequency. The volumetric absorption is only important for very large domains/rooms. It is furthermore assumed that maλ << 1 such that the diffusion coefficient remains unchanged.

The mean free path λ is the distance a sound particle on average travels between reflections. It is related to the average reflections frequency by λ = c/. The mean free path is a property of each room and for a regular cubic like room it can be calculated by (the usual convention)

where V is the room volume and S the total room surface area (see Ref. 18). The mean free path is in principle defined for every room as it depends on the room geometry and shape (see examples in Ref. 13 and Ref. 14). Entering a user defined mean free path based on for example measurements or a ray tracing simulation is also possible (see Ref. 1, Ref. 2, and Ref. 17).

In zones with fittings, like furniture or other absorbers, the scatterers are modeled statistically by their number density nf (SI unit: 1/m3), their average cross-section Qf (SI unit: m2), and their absorption coefficient αf (dimensionless). For the domain with scatterers the mean free path becomes

With these definitions in place, the governing diffusion equation can be modified to take the fitting into account (see, for example, Ref. 15)

The time limit after which the acoustic diffusion model leads to correct results has been discussed by many authors; see, for example, Ref. 6, Ref. 12, or Ref. 15. They suggest that a limit of one mean free time λ/c can be considered for the diffusion equation to have physical meaning. Before this time, the high probability of the particles of not having hit a scatterer or surface yet leads to invalid results. Typically, models of this type only apply at frequencies above the Schroeder frequency. This is a good “rule of thumb” measure — in several publications results have been seen to match measurements also at lower frequencies. Below the Schroeder frequency the room eigenmodes are important and can be modeled using pressure acoustics.

The source term q (x,t) (SI unit of power density J/m3/s = W/m3) can be defined as a point source or a volume source.

At walls, a mixed boundary condition accounting for absorption losses is used (the surface normal n being outward to the volume Ω)

where h is the exchange coefficient. Different models exist for this coefficient as given by Xiang and others (see Ref. 3 and Ref. 4). The Sabine type exchange coefficient is

This model is not suited for absorption coefficients close to 1. Finally, there is the (modified) model by Jing and Xiang (see Ref. 4):

On interior boundaries between rooms (thin walls, doors, grills, thin panels, and so on) there can be a transmission loss (TL) associated. The transmission loss is related to the transmission coefficient τ of the boundary as

This results in a condition where the field w is discontinuous across the boundary (a slit). The TL for an opening is typically given for the boundary in which the opening is located. That is, the combined transmission and absorption is smeared on a boundary.

The “room coupling” condition adds the fluxes related to the TL. Note also that the terms of the type τcw/4 should probably be modified for small TL because τ then becomes larger than 0.2 (at around 7 dB). An option exists to force continuity by constraining

This is in general the local SPL for any source. For a steady-state model, this is the spatial SPL distribution Lp(x).

The steady-state sound-energy decay d(x,t) can be calculated from an energy impulse response using so-called Schroeder integration. This corresponds to first finding the steady state (using a stationary solver) and then use this solution as initial condition (source turned off) in a time-dependent model. The same response can be calculated using Schroeder integration of an impulse response model

This integration can be set up in COMSOL Multiphysics postprocessing using the built-in timeint() operator.

A transient simulation is used to model the energy impulse response (the response to a source of the type E0λ(t), typically approximated by a step function of a short time) of a room or the steady-state energy decay response (decay from a steady state solution). In a single room, the response will be the same as measured in all points (except for a small time lag). This will result in identical reverberation times (RT) estimates. The interest of the transient model is more evident when several rooms are coupled. Here, several time scales exist because of the interaction between the rooms and these can be seen in a transient energy impulse response.

where λi is the i:th eigenvalue and wi the associated mode. The eigenvalues are a direct measures of the reverberation time (RT) of the different rooms. T60 is for example simply given by

where the last equation is the classical expression. Evaluating the eigenvalues will gives an easy measure of the slopes of the energy-time functions (ETFs) and can be combined with a transient model to give the full picture of the reverberation in the different volumes (rooms) and their coupled behavior. When performing an eigenvalue study, the variable ade.T60 gives the reverberation time associated with the given eigenvalue. Inspect the sound energy density modes plot to determine see to which room the reverberation time is associated.