2D Mesh and the Wave Equation

Figure C.35: Region of nodes in a rectilinear waveguide mesh.

Figure C.36: Zoom-in about node $(l,m)$ at time $n$ in a rectilinear waveguide mesh, showing traveling-wave components entering and leaving the node. (All variables are at time $n$ ,)

Consider the 2D rectilinear mesh, with nodes at positions $x=lX$ and $y=mY$ , where $l$ and $m$ are integers, and $X$ and $Y$ denote the spatial sampling intervals along $x$ and $y$ , respectively (see Fig.C.35). Then from Eq.(C.129) the junction velocity $v_{lm}$ at time $n$ is given by

$$v_{lm}(n) = \frac{1}{2}\left[ v_{lm}^{+\textsc{n}}(n) + v_{lm}^{+\textsc{e}}(n) + v_{lm}^{+\textsc{s}}(n) + v_{lm}^{+\textsc{w}}(n)\right]$$

where $v_{lm}^{+\textsc{n}}(n)$ is the ``incoming wave from the north'' to node $(l,m)$ , and similarly for the waves coming from east, west, and south (see Fig.C.36).

These incoming traveling-wave components arrive from the four neighboring nodes after a one-sample propagation delay. For example, $v_{lm}^{+\textsc{n}}(n)$ , arriving from the north, departed from node $(l,m+1)$ at time $n-1$ , as $v_{l,m+1}^{-\textsc{s}}(n-1)$ . Furthermore, the outgoing components at time $n$ will arrive at the neighboring nodes one sample in the future at time $n+1$ . For example, $v_{lm}^{-\textsc{n}}(n)$ will become $v_{l,m+1}^{+\textsc{s}}(n+1)$ . Using these relations, we can write $v_{lm}(n+1)$ in terms of the four outgoing waves from its neighbors at time $n$ :

$$v_{lm}(n+1) = \frac{1}{2}\left[ v_{l,m+1}^{-\textsc{s}}(n) + v_{l+1,m}^{-\textsc{w}}(n) + v_{l,m-1}^{-\textsc{n}}(n) + v_{l-1,m}^{-\textsc{e}}(n)\right] \protect$$ (C.137)

where, for instance, $v_{lm}^{-\textsc{n}}(n)$ is the ``outgoing wave to the north'' from node $(l,m)$ . Similarly, the outgoing waves leaving $v_{lm}(n-1)$ become the incoming traveling-wave components of its neighbors at time $n$ :

$$v_{lm}(n-1) = \frac{1}{2}\left[ v_{l,m+1}^{+\textsc{s}}(n) + v_{l+1,m}^{+\textsc{w}}(n) + v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right] \protect$$ (C.138)

This may be shown in detail by writing

$$\begin{eqnarray*} v_{lm}(n-1) &=& \frac{1}{2}[v_{lm}^{+\textsc{n}}(n-1) + \cdots + v_{lm}^{+\textsc{w}}(n-1)]\\ &=& \frac{1}{2}[v_{lm}(n-1) - v_{lm}^{-\textsc{n}}(n-1) + \cdots + v_{lm}(n-1) - v_{lm}^{-\textsc{w}}(n-1)]\\ &=& 2v_{lm}(n-1) - \frac{1}{2}\left[ v_{lm}^{-\textsc{n}}(n-1) + \cdots + v_{lm}^{-\textsc{w}}(n-1)\right] \end{eqnarray*}$$

so that

$$\begin{eqnarray*} v_{lm}(n-1) &=& \frac{1}{2}[v_{lm}^{-\textsc{n}}(n-1) + \cdots + v_{lm}^{-\textsc{w}}(n-1)]\\ &=& \frac{1}{2}\left[ v_{l,m+1}^{+\textsc{s}}(n) + v_{l+1,m}^{+\textsc{w}}(n) + v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right]. \end{eqnarray*}$$

Adding Equations (C.137-C.137), replacing $v_{l,m+1}^{+\textsc{s}}(n) + v_{l,m+1}^{-\textsc{s}}(n)$ with $v_{l,m+1}(n)$ , etc., yields a computation in terms of physical node velocities:

$$\begin{eqnarray*} \lefteqn{v_{lm}(n+1) + v_{lm}(n-1) = } \\ & & \frac{1}{2}\left[ v_{l,m+1}(n) + v_{l+1,m}(n) + v_{l,m-1}(n) + v_{l-1,m}(n)\right] \end{eqnarray*}$$

Thus, the rectangular waveguide mesh satisfies this equation giving a formula for the velocity at node $(l,m)$ , in terms of the velocity at its neighboring nodes one sample earlier, and itself two samples earlier. Subtracting $2v_{lm}(n)$ from both sides yields

$$\begin{eqnarray*} \lefteqn{v_{lm}(n+1) - 2 v_{lm}(n) + v_{lm}(n-1)} \\ &=& \frac{1}{2} \left\{ \left[v_{l,m+1}(n) - 2 v_{lm}(n) + v_{l,m-1}(n)\right] \right.\\ &&+ \left. \left[v_{l+1,m}(n) - 2 v_{lm}(n) + v_{l-1,m}(n)\right]\right\} \end{eqnarray*}$$

Dividing by the respective sampling intervals, and assuming $X=Y$ (square mesh-holes), we obtain

$$\begin{eqnarray*} \lefteqn{\frac{v_{lm}(n+1) - 2 v_{lm}(n) + v_{lm}(n-1)}{T^2}} \\ &=& \frac{X^2}{2T^2}\left[ \frac{v_{l,m+1}(n) - 2 v_{lm}(n) + v_{l,m-1}(n)}{Y^2} \right.\\ &&\quad\; +\left.\frac{v_{l+1,m}(n) - 2 v_{lm}(n) + v_{l-1,m}(n)}{X^2}\right]. \end{eqnarray*}$$

In the limit, as the sampling intervals $X,Y,T$ approach zero such that $X/T = Y/T$ remains constant, we recognize these expressions as the definitions of the partial derivatives with respect to $t$ , $x$ , and $y$ , respectively, yielding

$$\frac{\partial^2 v(t,x,y)}{\partial t^2} = \frac{X^2}{2T^2} \left[ \frac{\partial^2 v(t,x,y)}{\partial x^2} + \frac{\partial^2 v(t,x,y)}{\partial y^2} \right].$$

This final result is the ideal 2D wave equation $\ddot v = c^2 \nabla^2 v$ , i.e.,

$$\frac{\partial^2 v}{\partial t^2} = c^2 \left[ \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right]$$

with

$$c = \frac{1}{\sqrt{2}}\frac{X}{T} = \frac{\sqrt{2}}{2}\frac{X}{T}. \protect$$ (C.139)

Discussion regarding solving the 2D wave equation subject to boundary conditions appears in §B.8.3. Interpreting this value for the wave propagation speed $c$ , we see that every two time steps of $2T$ seconds corresponds to a spatial step of $\sqrt{2}X$ meters. This is the distance from one diagonal to the next in the square-hole mesh. We will show later that diagonal directions on the mesh support exact propagation (of plane waves traveling at 45-degree angles with respect to the $x$ or $y$ axes). In the $x$ and $y$ directions, propagation is highly dispersive, meaning that different frequencies travel at different speeds. The exactness of 45-degree angles can be appreciated by considering Huygens' principle on the mesh.