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Spatial Derivatives

Slope waves are simply related to velocity waves.
By the chain rule,

\begin{eqnarray*}
y'(t,x) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial x}y(t,x) \\ [5pt]
&=& y'_r(t-x/c) + y'_l(t+x/c)
\\ [5pt]
&=& -\frac{1}{c} {\dot y}_r(t-x/c) + \frac{1}{c}{\dot y}_l(t+x/c) \\ [5pt]
&\rightarrow& -\frac{1}{c} v^{+}(n-m) + \frac{1}{c}v^{-}(n+m)
\end{eqnarray*}

$ \Rightarrow$

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

or

\begin{displaymath}
\begin{array}{rcrl}
v^{+}&=&-&cy'^{+}\\ [5pt]
v^{-}&=&&cy'^{-}
\end{array}\end{displaymath}


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Download VariableChoice.pdf
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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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