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Use of the Chain Rule

These traveling-wave partial-derivative relations may be derived a bit more formally by means of the chain rule from calculus, which states that, for the composition of functions $ f$ and $ g$ , i.e.,

$\displaystyle y(x) = f(g(x)),
$

the derivative of the composition with respect to $ x$ can be expressed according to the chain rule as

$\displaystyle y'(x) = f^\prime(g(x))g^\prime(x),
$

where $ f^\prime(x)$ denotes the derivative of $ f(x)$ with respect to $ x$ .

To apply the chain rule to the spatial differentiation of traveling waves, define

\begin{eqnarray*}
g_r(t,x) &=& t - \frac{x}{c}\\ [10pt]
g_l(t,x) &=& t + \frac{x}{c}.
\end{eqnarray*}

Then the traveling-wave components can be written as $ y_r[g_r(t,x)]$ and $ y_l[g_l(t,x)]$ , and their partial derivatives with respect to $ x$ become

\begin{eqnarray*}
y'_r\;\isdef \; \frac{\partial}{\partial x} y_r\left[g_r(t,x)\right]
&=& \left(\left.\frac{\partial y_r(\xi)}{\partial \xi}\right\vert _{\xi=g_r(t,x)}
\right)\left( \left.\frac{\partial g_r(t,\xi)}{\partial \xi} \right\vert _{\xi=x}\right)\\ [10pt]
&=& {\dot y}_r[g_r(t,x)] \cdot \frac{\partial g_r(t,x)}{\partial x} \\ [10pt]
&=& {\dot y}_r[g_r(t,x)] \cdot \left(-\frac{1}{c}\right)
\;\isdef \; -\frac{1}{c}{\dot y}_r,
\end{eqnarray*}

and similarly for $ y'_l$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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