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.,

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

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

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

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 $y'_l$ .