Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

TL;DR

This paper establishes frequency-explicit bounds for acoustic wave transmission through layered obstacles with variable coefficients, and provides a homogenization analysis that is explicit in oscillation period and frequency, applicable to high-frequency regimes.

Contribution

It proves a uniform frequency-explicit bound for the Helmholtz equation with layered coefficients and derives a homogenization error estimate explicit in oscillation period and frequency.

Findings

Frequency-explicit bounds are valid for large frequencies and layered coefficients.

Homogenization error bounds are explicit in oscillation period and frequency.

Results apply to highly-oscillatory coefficients without small frequency assumptions.

Abstract

We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.