High Frequency Scattering by a Classically Invisible Body

TL;DR

This paper investigates the high-frequency scattering properties of a polyhedral object that is classically invisible, analyzing how wave interactions behave under various impedance boundary conditions and revealing oscillatory and vanishing cross-section phenomena.

Contribution

It provides a rigorous quasiclassical approximation for wave scattering by an invisible polyhedron and characterizes the asymptotic behavior of the total cross section at high frequencies.

Findings

Total momentum transmitted vanishes as frequency increases.

Total cross section oscillates at high frequencies.

Average cross section over impedance intervals tends to zero for certain frequencies.

Abstract

We consider a polyhedron with zero classical resistance, i.e., a polyhedron invisible to an observer viewing only the paths of geometrical optics rays. The corresponding problem of scattering of plane waves by the polyhedron is studied. The quasiclassical approximation is obtained and justified in the case of impedance boundary conditions with a non zero absorbing part. It is shown that the total momentum transmitted to the obstacle vanishes when the frequency $k$ goes to infinity, and that the total cross section oscillates at high frequencies. When the impedance $\lambda_0$ is real (i. e., there is no absorption), it is shown that there exists a sequence of frequencies $k_n$ such that the averages in the impedance of the total cross section over shrinking intervals around $\lambda_0 $ go to zero as $k_n \to \infty$.