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.1Then a general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as
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point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units.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 . See Appendix A for more on the history of the wave equation and related topics.