In this paper, the problem of wave propagation in periodic structured composites is
studied, and a dispersive asymptotic method for the description of these dynamic
processes is proposed. Assuming a single-frequency dependence of the solution for the
one dimensional wave equation in a periodic composite material, higher-order terms
in the asymptotic expansion for the displacement functions are studied.
Nonuniformity is eliminated by finding a suitable regular asymptotic expansion for
the perturbation frequency. Only two spatial scales are considered, and the
equivalence of this method and the introduction of multiple slow temporal scales
is shown, in good agreement with previous approaches. For a selection of
boundary problems, analytic solutions are given and graphically illustrated. The
problem of failures is also discussed, and some illustrative calculations are
presented.
Instituto de Investigaciones en
Matemáticas Aplicadas y en Sistemas
Universidad Nacional Autónoma de México
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Mexico