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Traveling-Wave Solution

It is easily shown that the lossless 1D wave equation $ Ky''=\epsilon {\ddot y}$ is solved by any string shape which travels to the left or right with speed $ c \isdeftext \sqrt{K/\epsilon }$ . 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 assumed twice-differentiable.C.1Then a general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

$\displaystyle y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right). \protect$ (C.11)

The next section derives the result that $ {\ddot y}_r= c^2y''_r$ and $ {\ddot y}_l= c^2y''_l$ , establishing that the wave equation is satisfied for all traveling wave shapes $ y_r$ and $ y_l$ . However, remember that the derivation of the wave equation in §B.6 assumes the string slope is much less than $ 1$ at all times and positions. Finally, we show in §C.3.6 that the traveling-wave picture is general; that is, any physical state of the string can be converted to a pair of equivalent traveling force- or velocity-wave components.

An important point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables $ y(t,x)$ has been replaced by two functions of a single variable in time units. This leads to great reductions in computational complexity.

The traveling-wave solution of the wave equation was first published by d'Alembert in 1747 [100]. See Appendix A for more on the history of the wave equation and related topics.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University