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Equivalent Finite Difference Scheme

We have

$\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]
$

$\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]
$

Adding gives a finite difference equation satisfied by the mesh

$\displaystyle \zbox{v_{lm}(n+1) +
v_{lm}(n-1) = \frac{v_{l,m+1} + v_{l+1,m} + v_{l,m-1} + v_{l-1,m}}{2}}
$

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*}

or, assuming $ X=Y$ (``square hole'' case),

\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,

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

i.e., the ideal 2D wave equation

$\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] \mathrel{\stackrel{\mathrm{\Delta}}{=}}c^2\nabla ^2 v
$

where $ \nabla ^2$ denotes the Laplacian, and

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


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``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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