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

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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.