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The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 . [ABSTRACT FROM AUTHOR]

We consider the wave equation in an unbounded domain and are interested in domain truncation methods. Our aim is to develop a numerical scheme that allows calculations for truncated waveguide geometries with periodic coefficient functions. The scheme is constructed with radiation boxes that are attached to the artificially introduced boundaries. A Dirichlet-to-Neumann operator |$N$| is calculated in these radiation boxes. Efficiency of the scheme is obtained by calculating |$N$| not with an iteration, but with a single run through the time interval. We observe speed-up factors of up to |$20$| in comparison to calculations without domain truncation. [ABSTRACT FROM AUTHOR]

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix aε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions uε well by a non-dispersive wave equation on fixed time intervals (0,T). Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions wε describe uε well on time intervals (0,Tε-2). More precisely, we provide a norm and uniform error estimates of the form uε;(t) -wε (t)≤ Cε for t ∈(0,Tε-2). They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an ε-independent equation of third order that describes dispersion along rays and we present numerical examples. [ABSTRACT FROM AUTHOR]

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in Rn, n ∊ {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length ε > 0, an effective second order wave equation that describes solutions on time intervals [0, T]. In order to approximate solutions on large time intervals [0, Tε-2], one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests. [ABSTRACT FROM AUTHOR]