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FDA of 1D Wave Equation

Substituting finite difference approximation (FDA) into the wave equation $ Ky''= \epsilon {\ddot y}$ gives

\begin{eqnarray*}
& & \\
& & K\frac{y(t,x+X) - 2 y(t,x) + y(t,x-X)}{X^2} \\
& & \\
& = & \epsilon \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x)}{T^2}
\end{eqnarray*}

$ \Rightarrow$ Time Update:

\begin{eqnarray*}
y(t+T,x) & = & \frac{KT^2}{\epsilon X^2}
\left[ y(t,x+X) - 2...
...)\right] \\
& & \qquad\qquad\qquad\qquad + 2 y(t,x) - y(t-T,x)
\end{eqnarray*}

Let $ c\mathrel{\stackrel{\Delta}{=}}\sqrt{K/\epsilon }$ (speed of sound along the string).
In practice, we typically normalize such that


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``Distributed Modeling in Discrete Time'', by Julius O. Smith III and Nelson Lee,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2008-06-05 under the Creative Commons License (Attribution 2.5), by Julius O. Smith III and Nelson Lee
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA