Acoustic Scattering from Corners, Edges and Circular Cones

TL;DR

This paper proves that obstacles with corners, edges, or circular conic points in acoustic scattering always produce a detectable scattered wave, and it establishes uniqueness in shape recovery from a single wave.

Contribution

It introduces a novel analysis of acoustic scattering from obstacles with conic points and edges, proving non-trivial scattering and uniqueness in inverse shape problems.

Findings

Obstacles with conic points or edges always scatter incoming waves non-trivially.

Unique shape reconstruction is possible from a single incoming wave.

The approach uses singularity analysis of the Laplace equation in cones.

Abstract

Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be any number in $(0,2\pi)\backslash\{\pi\}$. We prove that such an obstacle scatters any incoming wave non-trivially (i.e., the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of a penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone.