Traveling-Wave Solution

It is easily shown that the lossless 1D wave equation $Ky''=\epsilon {\ddot y}$ is solved by any string shape which travels to the left or right with speed $c \isdeftext \sqrt{K/\epsilon }$. Denote right-going traveling waves in general by $y_r(t-x/c)$ and left-going traveling waves by $y_l(t+x/c)$, where $y_r$ and $y_l$ are assumed twice-differentiable.^G.1Then a general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

$$y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right). \protect$$ (G.11)

The next section derives the result that ${\ddot y}_r= c^2y''_r$ and ${\ddot y}_l= c^2y''_l$, establishing that the wave equation is satisfied for all traveling wave shapes $y_r$ and $y_l$. However, remember that the derivation of the wave equation in Appendix F assumes the string slope is much less than $1$ at all times and positions. Finally, we show in §G.3.5 that the traveling-wave picture is general; that is, any physical state of the string can be converted to a pair of equivalent traveling force- or velocity-wave components.

An important point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables $y(t,x)$ has been replaced by two functions of a single variable in time units. This leads to great reductions in computational complexity.

The traveling-wave solution of the wave equation was first published by d'Alembert in 1747 [93]. See Appendix D for more on the history of the wave equation and related topics.