Asymptotic solutions of the Cauchy problem for a wave equation with rapidly varying coefficients

For the wave equation in which the squared wave propagation velocity is a small rapidly oscillating perturbation of a slowly varying function, we consider the Cauchy problem with initial data localized in a small neighborhood of some point. Assuming that the perturbation lies in some algebra of averageable functions and the small parameters characterizing the localization of the initial data and the oscillation rate and amplitude of the perturbation are related by certain inequalities, we show that the leading term of the asymptotics of the solution can be obtained by the replacement of the velocity with its local average. We discuss classes of averageable functions, the relationship between our approach and other approaches to homogenization, and possible applications to models of tsunami wave propagation.
This work was done together with S. Dobrokhotov and B. Tirozzi and was supported by RFBR grant 14-01-00521 and by the CINFAI-RITMARE project (Italy).