On the scattered field generated by a ball inhomogeneity of constant index

TL;DR

This paper derives precise estimates for the scattered field caused by a small inhomogeneity in the index within a scalar Helmholtz framework, providing broadband bounds valid across frequencies and contrasts.

Contribution

It introduces sharp estimates and a broadband uniform bound for the scattered field generated by a disk inhomogeneity of constant index in the Helmholtz equation.

Findings

Sharp estimates of scattered field size for all frequencies and contrasts.

A broadband uniform bound valid outside a vanishing frequency set.

Results applicable to small inhomogeneities in wave scattering scenarios.

Abstract

We consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk of radius $\epsilon$ and another one outside. We derive sharp estimates of the size of the scattered field caused by this disk inhomogeneity, for any frequencies and any contrast. We also provide a broadband estimate, that is, a uniform bound for the scattered field for any contrast, and any frequencies outside of a set which tend to zero with $\epsilon$.