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

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

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

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,

and similarly for $ y'_l$.

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