Dispersive effective models for waves in heterogeneous media
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Date
2010-03-23T14:45:40Z
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Abstract
We study the long time behavior of waves in a strongly heterogeneous
medium, starting from the one-dimensional scalar wave equation with
variable coefficients. We assume that the coefficients are periodic with period
ɛ and ɛ > 0 is a small length parameter. Our main result is the rigorous
derivation of two different dispersive models. The first is a fourth-order
equation with constant coefficients including powers of ɛ . In the second
model, the ɛ-dependence is completely avoided by considering a third-order
linearized Korteweg-de-Vries equation. Our result is that both simplified
models describe the long time behavior well. An essential tool in our analysis
is an adaption operator which modifies smooth functions according to the
periodic structure of the medium.
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Keywords
homogenization, wave equation, dispersive model, long time behavior