Introduction

In this appendix from [2], we reexamine the finite difference schemes corresponding to waveguide meshes discussed in Chapter 4 of [2] in the special case for which the underlying model problem is lossless, source-free, and does not exhibit any material parameter variation. In this case, these finite difference schemes will solve the wave equation, given by

$$\frac{\partial^{2} u}{\partial t^{2}} = \gamma^{2}\nabla^{2}u$$ (1)

in either (2+1)D or (3+1)D, depending on the type of mesh. Here, $\gamma$ is the wave speed, and $\nabla^{2}$ is the Laplacian [6]. These schemes will be linear and shift-invariant, and as such, it is possible to analyze them in the frequency domain, through what is called Von Neumann analysis [8]. We will apply these methods to the rectilinear, interpolated rectilinear, triangular, hexagonal and fourth-order accurate schemes in (2+1)D, then to the cubic rectilinear, interpolated cubic rectilinear, octahedral and tetrahedral schemes in (3+1)D.