Homogenization of the variable-speed wave equation

The existence of traveling waves in strongly inhomogeneous media is discussed in the framework of the one-dimensional linear wave equation with a variable speed. Such solutions are found by using a homogenization, when the variable-coefficient wave equation transforms 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. Some particular solutions recover asymptotic WKB approximations for slowly varying waves in inhomogeneous media obtained earlier in the literature. We also show that the wave equation under the same constraint on the variable speed admits a two-parameter Lie group of nontrivial commuting point symmetries.