Leading and second order homogenization of an elastic scattering problem for highly oscillating anisotropic medium

TL;DR

This paper develops a two-scale and higher-order homogenization approach to analyze elastic wave scattering in highly oscillating anisotropic media, deriving effective equations and dispersion properties.

Contribution

It introduces a second-order homogenized model and a fourth-order dispersion model for elastic waves in complex anisotropic media, advancing the understanding of wave behavior in such structures.

Findings

Derived a constant coefficient second-order PDE for effective wave propagation.

Established a fourth-order PDE capturing anisotropic dispersion effects.

Analyzed convergence rates using boundary correctors.

Abstract

We consider the scattering of elastic waves by highly oscillating anisotropic periodic media with bounded support. Applying the two-scale homogenization, we first obtain a constant coefficient second-order partial differential elliptic equation that describes the wave propagation of the effective or overall wave field. We study the rate of convergence by introducing complimentary boundary correctors. To account for dispersion induced by the periodic structure, we further pursue a higher-order homogenization. We then investigate the rate of convergence and formally obtain a fourth-order differential equation that demonstrates the anisotropic dispersion