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

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

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]

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