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Ohm's Law for Traveling Waves

We just derived Ohm's Law for Traveling Waves on an Ideal String

\fbox{%
\begin{minipage}[c]{3in}%
\begin{displaymath}\begin{array}{rcrl}%
f^{{+}}(n)&=&&R\,v^{+}(n) \\
f^{{-}}(n)&=&-&R\,v^{-}(n)
\end{array}\end{displaymath}\end{minipage}}
where the velocity waves are defined in terms of transverse string displacement by

\begin{eqnarray*}
v^{+}(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \dot y^{+}(n)\\
v^{-}(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \dot y^{-}(n),
\end{eqnarray*}

$ f^{{+}}$ and $ f^{{-}}$ are corresponding force waves, and $ R\mathrel{\stackrel{\mathrm{\Delta}}{=}}\sqrt{K\epsilon }$ is the wave impedance of the string:

$\displaystyle \zbox{R \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\sqrt{K\epsilon } \;=\;\frac{K}{c} \;=\;\epsilon c}
$


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``Choice of Wave Variables in Digital Waveguide Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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