Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions

TL;DR

This paper studies acoustic scattering by penetrable objects, identifying conditions for non-scattering wavenumbers and constructing invisible inclusions that cannot be detected from limited far field measurements, with practical algorithms.

Contribution

It establishes the discreteness of non-scattering wavenumbers and provides a constructive method to design invisible inclusions for specific frequencies and domains.

Findings

Non-scattering wavenumbers form a discrete set under certain conditions.

Existence of inclusions that are undetectable from limited far field data.

A numerical algorithm for approximating invisible inclusions.

Abstract

We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain D, we present a constructive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which is described at the end of the paper and which allows to provide examples of (approximated) invisible inclusions.