Homogenization of the variable-speed wave equation

The existence of traveling waves in strongly inhomogeneous media is reviewed in the framework of the one-dimensional linear wave equation with a variable speed. Such solutions are found by using a homogenization, in which the variable-coefficient wave equation is transformed to a constant-coefficient Klein–Gordon equation. This transformation exists if and only if the spatial variations of the variable speed satisfy a constraint expressed by a second-order ordinary differential equation with two arbitrary parameters. All solutions of the constraint are found in explicit form, and our results obtained by this systematic procedure include many previous results found in the literature. Further, we show that the wave equation under the same constraint on the variable speed admits a two-parameter Lie group of nontrivial commuting point symmetries.