Use of the Chain Rule

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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 and , i.e.,

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

the derivative of the composition with respect to

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

with respect to

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 and , and their partial derivatives with respect to become

$\begin{eqnarray*} y'_r\;\isdef \; \frac{\partial}{\partial x} y_r\left[g_r(t,x)\... ...t \left(-\frac{1}{c}\right) \;\isdef \; -\frac{1}{c}{\dot y}_r, \end{eqnarray*}$

and similarly for .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Physical Audio Signal Processing
    Digital Waveguide Theory
       Traveling-Wave Solution
          Use of the Chain Rule