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Wave Impedance

We just showed

\begin{displaymath}
\begin{array}{rcrl}
y'^{+}&=&-&\frac{1}{c}v^{+}\\ [5pt]
y'^{-}&=&&\frac{1}{c}v^{-}
\end{array}\end{displaymath}

Define new wave variables in terms of slope waves as

\begin{eqnarray*}
f^{{+}}&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& - Ky'^{+}\\
f^{{-}}&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& - Ky'^{-}
\end{eqnarray*}

Note that $ f^\pm $ are in physical units of force.
We have

\begin{displaymath}
\begin{array}{rcrl}
f^{{+}}&=&&\frac{K}{c}v^{+}\\ [5pt]
f^{{-}}&=&-&\frac{K}{c}v^{-}
\end{array}\end{displaymath}

Recall

\begin{eqnarray*}
c &=& \sqrt{\frac{K}{\epsilon }}\\ [10pt]
\Rightarrow\qquad
\frac{K}{c} &=& \sqrt{K\epsilon } \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\; R
\end{eqnarray*}

which is the wave impedance of the ideal string (force/velocity for traveling waves). Thus,

\begin{displaymath}
\zbox{
\begin{array}{rcrl}
f^{{+}}&=&&R\,v^{+}\\
f^{{-}}&=&-&R\,v^{-}
\end{array}}
\end{displaymath}


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Download VariableChoice.pdf
Download VariableChoice_2up.pdf
Download VariableChoice_4up.pdf

``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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