Cones with convoluted geometry that always scatter or radiate

TL;DR

This paper demonstrates that certain conical potentials with irregular geometries always scatter incident waves at fixed energy, showing they are never transparent, and uses spherical harmonics and deformation methods for analysis.

Contribution

It introduces a class of irregular conical scatterers that always produce a scattered wave, expanding understanding of wave scattering by complex geometries.

Findings

Large class of cones always scatter incident waves

Most deformations between circular and star-shaped cones always scatter

Sources supported on thin cones produce non-zero far-field

Abstract

We investigate fixed energy scattering from conical potentials having an irregular cross-section. The incident wave can be any arbitrary non-trivial Herglotz wave. We show that a large number of such local conical scatterers scatter all incident waves, meaning that the far-field will always be non-zero. In essence there are no incident waves for which these potentials would seem transparent at any given energy. We show more specifically that there is a large collection of star-shaped cones whose local geometries always produce a scattered wave. In fact, except for a countable set, all cones from a family of deformations between a circular and a star-shaped cone will always scatter any non-trivial incident Herglotz wave. Our methods are based on the use of spherical harmonics and a deformation argument. We also investigate the related problem for sources. In particular if the support of the source is locally a thin cone, with an arbitrary cross-section, then it will produce a non-zero far-field.