Two-Dimensional Wave Propagation in Layered Periodic Media

Abstract

We study two-dimensional wave propagation in materials whose properties vary periodically in one direction only. High order homogenization is carried out to derive a dispersive effective medium approximation. One-dimensional materials with constant impedance exhibit no effective dispersion. We show that a new kind of effective dispersion may arise in two dimensions, even in materials with constant impedance. This dispersion is a macroscopic effect of microscopic diffraction caused by spatial variation in the sound speed. We analyze this dispersive effect by using highorder homogenization to derive an anisotropic, dispersive effective medium. We generalize to two dimensions a homogenization approach that has been used previously for one-dimensional problems. Pseudospectral solutions of the effective medium equations agree to high accuracy with finite volume direct numerical simulations of the variable-coefficient equations.

Diffractons: Solitary Waves Created by Diffraction in Periodic Media

“…Numerically integrating (13) with an appropriate velocity V yields solitary waves nearly identical to the homogenized diffractons of Figure 4.…”

“…On the other hand, if the linearized sound speed is constant then all the dispersive term coefficients vanish. This is because the effective dispersive mechanism in this case is that of diffraction, which occurs only if the sound speeds differ [13]. In the absence of diffraction, nonlinearity leads to shock formation, as observed in Figure 3(c).…”

“…In [13], high order homogenized equations are derived for linear acoustic waves in a medium of the type considered in this work. It is assumed that λ, the typical wavelength of the solution, is large compared to Ω, the period of the medium, so that δ = Ω/λ is a small parameter.…”

“…The homogenized equations presented in Section 3 are derived through a nonlinear extension of the work in [13] applied to system (3) with the constitutive relation (2). Here we work through the derivation; the reader is refered to [13] for even more detail.…”

“…We emphasize that this effective dispersion is a macroscopic effect of the material microstructure; clearly, no dispersive terms appear in the model equations. This effect has been studied in detail for linear waves in [13]. Next we consider the same two scenarios, but with the nonlinear stress relation (2).…”