Traveling-Wave Solution

It is easily shown that the lossless 1D wave equation
is solved by any string shape which travels to the left or right with
speed
. Denote *right-going*
traveling waves in general by
and *left-going*
traveling waves by
, where
and
are assumed
twice-differentiable.^{C.1}Then a general class of solutions to the
lossless, one-dimensional, second-order wave equation can be expressed
as

The next section derives the result that and , establishing that the wave equation is satisfied for all traveling wave shapes and . However, remember that the derivation of the wave equation in §B.6 assumes the string slope is much less than at all times and positions. Finally, we show in §C.3.6 that the traveling-wave picture is

An important point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units.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.

- Traveling-Wave Partial Derivatives
- Use of the Chain Rule
- String Slope from Velocity Waves
- Wave Velocity
- D'Alembert Derived
- Converting String-State to Traveling-Waves

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University