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.^H.1Then 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 Appendix G assumes the string slope is much less than at all times and positions. Finally, we show in §H.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point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units.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 [99]. See Appendix E for more on the history of the wave equation and related topics.

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Subsections

• 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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written by 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.