Physical Room Modeling with DWNs

Recent developments in physical modeling using digital waveguides have included the use of a waveguide mesh to model 2D membranes and 3D rooms [32,33,24]. In the membrane, for example, a rectilinear mesh of digital waveguides can be interconnected via four-port scattering junctions to provide lossless prototypes for ``plate reverberators'' and the like. A single dispersive waveguide (made dispersive using embedded allpass filters) can be used to model ``spring reverberators.'' Savioja et al. [24] have found that the rectilinear 3D waveguide mesh has good room simulation properties at low frequencies.

Since reverberation quality generally increases with the number dimensions (from spring to plate to acoustic space), it is plausible to expect that higher dimensional waveguide meshes will provide better reverberation than we have ever known. Generalizing (35) to higher dimensions, one can see that the higher the dimensionality, the more rapidly the mode density increases with frequency. However, above the ``Schroeder limit'' at which the modes are so densely packed that the ear cannot resolve them, increasing the density should have no audible effect. Nevertheless, it is an interesting direction to pursue.

The waveguide mesh is structurally lossless so that there is no attenuation error in the sampled wave propagation. However, the grid quantization does give rise to dispersion error: The speed of sound effectively varies somewhat as a function of frequency and propagation direction on the mesh. Generally, results are very accurate at low frequencies, but sound speed decreases gradually as frequency increases in all but certain directions which tend to be diagonals along the mesh [32]. The choice of mesh geometry has a strong effect on the dispersion behavior [33]. It also strongly affects computational complexity. As an example, whenever an isotropic mesh utilizes $N$-port scattering junctions in which $N$ is a power of 2, the scattering matrices require no multiplies [26]. For rectilinear meshes, membranes are multiply free, as are solids in 4D (since the number of ports is $2N$, in $N$ dimensional space). The tetrahedral mesh, analogous to the diamond crystal, requires no multiplies to fill 3D space. Multiply-free waveguide meshes can be integrated very densely in VLSI.

A final word about waveguide meshes is that they, like any other LTI system, can be expressed in a sparse state-space form which yields an FDN that can be interpreted as a physical model.