### Traveling-Wave Solution

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

*any string shape*which travels to the left or right with speed

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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Sampled Traveling-Wave Solution

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Wave Equation Applications