Signal-Theoretic Characterization of Waveguide Mesh Geometries for Models of Two--Dimensional Wave Propagation in Elastic Media

TL;DR

This paper uses multidimensional sampling theory to compare the square, triangular, and hexagonal meshes in terms of sampling efficiency and dispersion error under conditions of critical sampling and shows that the triangular geometry exhibits the most desirable tradeoff between accuracy and computational cost.

Abstract

Waveguide Meshes are efficient and versatile models of wave propagation along a multidimensional ideal medium. The choice of the mesh geometry affects both the computational cost and the accuracy of simulations. In this paper, we focus on 2D geometries and use multidimensional sampling theory to compare the square, triangular, and hexagonal meshes in terms of sampling efficiency and dispersion error under conditions of critical sampling. The analysis shows that the triangular geometry exhibits the most desirable tradeoff between accuracy and computational cost.

Figures (6)

• Figure 1: The SWM (a), the TWM (b) and and the HWM (c). $-$,$/$,$\setminus$ and $|$ are digital waveguides, $\bullet$ are lossless scattering junctions. In (c), seven lossless scattering junctions separated by two digital waveguide branches are marked with $\odot$.
• Figure 2: Propagation speed ratios in the SWM (a), TWM (b) and HWM (c), versus domain given by (\ref{['StartingDomain']}).
• Figure 3: The HWM, obtained by subtraction of TWMs. $\times$ are elements belonging to $L_{\hbox{\small T}}(D_{\hbox{\small h}})$, $\bullet$ are elements belonging to $L_{\hbox{\small t}}(D_{\hbox{\small h}})$.
• Figure 4: Domains of Fourier images in a SWM (empty circles in dashed line) and in a TWM (filled circles), and the correspondent tiling induced by the respective geometries (square in dashed line and hexagon in solid line). The empty and the filled circles have the same radius and touch each other without intersecting, meaning that both the SWM and the TWM critically sample the same signal.
• Figure 5: Centers ($\times$) of the Fourier images coming from a TWM defined by $L_{\hbox{\small t}}(D_{\hbox{\small h}})$, and centers ($\bullet$) of the Fourier images coming from a sparser TWM defined by $L_{\hbox{\small T}}(D_{\hbox{\small h}})$. Subtraction of $\times$ from $\bullet$ gives the centers of the Fourier images (located on the filled circles) coming from an HWM defined by $L_{\hbox{\small h}}(D_{\hbox{\small h}})$. The respective tiling is given by the triangles. Both the HWM and the sparser TWM critically sample the same signal.