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String Wave Equation Derivation

\begin{figure}\centering
\input fig/wed.pstex_t
\\ {\LARGE Force diagram for length $dx$\ string element}
\end{figure}


Total upward force on length $ dx$ string element:

\begin{eqnarray*}
f(x+dx/2) &=& K\sin(\theta_1) + K\sin(\theta_2)\\
&\approx& K...
...\approx& K\left[-y'(x) + y'(x)+y''(x)dx)\right]\\
&=& Ky''(x)dx
\end{eqnarray*}

Mass of length $ dx$ string segment: $ m=\epsilon \,dx$.

By Newton's law, $ f=ma=m{\ddot y}$, we have

$\displaystyle Ky''(t,x)dx = (\epsilon \,dx){\ddot y}(t,x)
$

or

$\displaystyle \zbox{Ky''(t,x) = \epsilon {\ddot y}(t,x)}
$


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``Elementary Digital Waveguide Models for Vibrating Strings'', 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