The Tetrahedral Digital Waveguide Mesh

The 2D digital waveguide mesh [7, 8] has proven to be e ective and e cient in the modeling of musical membranes and plates, particularly in combination with recent simpli cations in modeling sti ness [6], nonlinearities [5], and felt mallet excitations [5]. The rectilinear 3D extension to the mesh had been suggested [8], and has been applied to the case of room acoustics [2]. However, it requires the use of 6-port scattering junctions, which make a multiply-free implementation impossible in the isotropic case. The 4-port scattering junctions of the 2D mesh required only an internal divide by 2, which could be implemented as a right shift in binary arithmetic. However, the 6port junction requires a divide by 3. The multiplyfree cases occur for N -port junctions in which N is a power of two [3]. We propose here a tetrahedral distribution of multiply-free 4-port scattering junctions lling space much like the molecular structure of the diamond crystal, where the placement of the scattering junctions corresponds to the placement of the carbon nuclei, and the bi-directional delay units correspond to the four tetrahedrally spaced single bonds between each pair of nuclei. We show that the tetrahedral mesh is mathematically equivalent to a nite di erence scheme (FDS) which approximates the 3D lossless wave equation. We further compute the frequencyand direction-dependent plane wave propagation speed dispersion error. 1 FDS View of the Mesh Rectilinear meshes compute an FDS approximation of the lossless wave equation [2, 8]. It is less obvious in the tetrahedral case. Figure 1 shows a small chunk of the tetrahedral mesh. We take the distance between adjacent junctions to be 1, and the junction point marked A to lie at the origin of an (x; y; z) cartesian coordinate system. We arrange the junctions B(0; 2 p 2=3; 1=3), C( p 2=3; p 2=3; 1=3), A B C