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R. Grimshaw, D. Pelinovsky, and E. Pelinovsky

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Homogenization of the variable-speed wave equation

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Wave Motion 47, 496-507 (2010)

**Abstract:**

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.

**Keywords:**

Wave equation, Klein-Gordon equation, point symmetries, traveling waves, transformations.