Traveling-Wave Solution

One-dimensional lossless wave equation:

$$\zbox{ Ky''= \epsilon {\ddot y}}$$

Plug in traveling wave to the right:

$$\begin{eqnarray*} y(t,x) &=& y_r(t-x/c) \\ & & \\ \Rightarrow \quad y'(t,x) &=& -\frac{1}{c}{\dot y}(t,x) \\ y''(t,x) &=& \;\;\;\frac{1}{c^2}{\ddot y}(t,x) \end{eqnarray*}$$

• Given $c\mathrel{\stackrel{\mathrm{\Delta}}{=}}\sqrt{K/\epsilon }$ , the wave equation is satisfied for any shape traveling to the right at speed $c$ (but remember slope $\ll 1$ )

• Similarly, any left-going traveling wave at speed $c$ , $y_l(t+x/c)$ , satisfies the wave equation (show)

• General solution to lossless, 1D, second-order wave equation:
  $$y(t,x) = y_r(t-x/c) + y_l(t+x/c)$$

• $y_l(\cdot)$ and $y_r(\cdot)$ are arbitrary twice-differentiable functions (slope $\ll 1$ )

• Important point: Function of two variables $y(t,x)$ is replaced by two functions of a single (time) variable $\Rightarrow$ reduced computational complexity.

• Published by d'Alembert in 1747
  (wave equation itself introduced in same paper)