Traveling Waves on the 2D Square-Holed Mesh

We found that the 2D digital waveguide mesh satisfies a finite difference scheme which converges to the ideal 2D wave equation with wave propagation speed

$$c = \frac{1}{\sqrt{2}}\frac{X}{T} = \frac{\sqrt{2}X}{2T}$$

• Every two time steps ($2T$ sec) corresponds to a spatial step of $\sqrt{2}X$ meters -- This is the distance from one diagonal to the next on the square-holed mesh
• Diagonal plane-wave propagation is exact
• Consider Huygens' principle along a mesh diagonal
• The $x$ and $y$ directions are highly dispersive:
  • High frequencies travel slower than low frequencies
  • Dispersion depends on frequency and direction
• The triangular mesh is much closer to isotropic:
  • Dispersion more nearly the same in all directions
• Frequency-dependent dispersion can be addressed using frequency warping
• By construction, there is no attenuation at any frequency in any direction