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String Wave Equation Derivation

\epsfig{file=eps/wed.eps}
Force diagram for length $ dx$ string element


Total upward force on length $ dx$ string element:

\begin{eqnarray*}
f(x+dx/2) &=& K\sin(\theta_1) + K\sin(\theta_2)\\
&\approx& K\left[\tan(\theta_1) + \tan(\theta_2)\right]\\
&=& K\left[-y'(x) + y'(x+dx)\right]\\
&\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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Download StringWaves.pdf
Download StringWaves_2up.pdf
Download StringWaves_4up.pdf

``Digitizing Strings Waves in Vibrating Strings'', 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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