A dispersive nonlocal model for wave propagation in periodic composites

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 singlefrequency 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.