Traveling-Wave Solution

It can be readily checked (see §C.3 for details) that the lossless 1D wave equation

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

(where all terms are defined in Eq.(6.1)) is solved by any string shape which travels to the left or right with speed

$$c \isdeftext \sqrt{\frac{K}{\epsilon }}.$$

If we 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 arbitrary twice-differentiable functions, then the general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

$$y(t,x) = y_r(t-x/c) + y_l(t+x/c). \protect$$ (7.2)

Note that we have ${\ddot y}_r= c^2y''_r$ and ${\ddot y}_l= c^2y''_l$ (derived in §C.3.1) showing that the wave equation is satisfied for all traveling wave shapes $y_r$ and $y_l$ . However, the derivation of the wave equation itself assumes the string slope $\vert y^\prime\vert$ is much less than $1$ at all times and positions (see §B.6). An important point to note is that a function of two variables $y(t,x)$ is replaced by two functions of a single (time) variable. This leads to great reductions in computational complexity, as we will see. The traveling-wave solution of the wave equation was first published by d'Alembert in 1747 [100]^7.1