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2D Mesh and the Wave Equation

Figure C.33: Region of nodes in a rectilinear waveguide mesh.
\includegraphics{eps/mesh}

Figure C.34: 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$ ,)
\includegraphics{eps/meshzoom}

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.33). Then from Eq.$ \,$ (C.105) the junction velocity $ v_{lm}$ at time $ n$ is given by

$\displaystyle 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.34).

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$ :

$\displaystyle v_{lm}(n+1) = \frac{1}{2}\left[ v_{l,m+1}^{-\textsc{s}}(n) + v_{l+1,m}^{-\textsc{w}}(n) + v_{l,m-1}^{-\textsc{n}}(n) + v_{l-1,m}^{-\textsc{e}}(n)\right] \protect$ (C.113)

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$ :

$\displaystyle v_{lm}(n-1) = \frac{1}{2}\left[ v_{l,m+1}^{+\textsc{s}}(n) + v_{l+1,m}^{+\textsc{w}}(n) + v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right] \protect$ (C.114)

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.113-C.113), 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

$\displaystyle \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.,

$\displaystyle \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

$\displaystyle c = \frac{1}{\sqrt{2}}\frac{X}{T} = \frac{\sqrt{2}}{2}\frac{X}{T}. \protect$ (C.115)

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.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA