In an ideal membrane, the speed of wave travel is independent of direction and spatial frequency (i.e., there is no wave dispersion). In a time interval T, all wave components will advance in position by cT.
The propagation distances between junctions on a 2D rectilinear grid, however, are not spatially uniform in all directions. As a result, wave propagation on the grid varies with direction as well as spatial frequency.
Mesh wave speed vs. spatial frequency.
Figure 3 is a plot of normalized wave speed versus spatial frequency and direction. The center region corresponds to low spatial frequencies and outer regions correspond to higher spatial frequencies. Angular direction, as seen from the frequency-plane origin, corresponds exactly to the direction of travel on the mesh.
No dispersion occurs for those frequencies and directions where the normalized wave speed equals 1 in Fig. 3. Thus, wave motion at low frequencies and along diagonal directions to the mesh coordinate system travel without dispersion error.
Waves traveling along the coordinate axes of the mesh, however, have wave speeds that fall off in proportion to frequency.
In a bounded mesh, dispersion errors result in mistunings of resonant modes.