### Traveling-Wave Solution

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

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

If we denote right-going traveling waves in general by and left-going traveling waves by , where and are arbitrary twice-differentiable functions, then the general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

 (7.2)

Note that we have and (derived in §C.3.1) showing that the wave equation is satisfied for all traveling wave shapes and . However, the derivation of the wave equation itself assumes the string slope is much less than at all times and positions (see §B.6). An important point to note is that a function of two variables 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

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