Unique determination of balls and polyhedral scatterers with a single point source wave

Abstract

In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer in the inverse acoustic scattering problem with a single incident point source wave in  N (N = 2,3). Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane or sphere, which is essentially a 'point-to-point' extension formula. The method has been adapted to proving uniqueness in inverse scattering from sound-soft cavities with interior measurement data deriving from a single point source. The corresponding uniqueness for sound-hard balls or polyhedral scatterers has also been discussed.

Uniqueness to inverse acoustic and electromagnetic scattering from locally perturbed rough surfaces

“…Repeating the reflection and path arguments used in [13,14], one can always find a desired flat surface whose maximum extension satisfies the above condition.…”

“…The aim of this paper is to prove the unique determination of polyhedral-type perfect conductors D by a single electric dipole with a fixed wave number, a fixed dipole point and a fixed polarization. The main tools we have used are the reflection principle for the Maxwell equations [24] and the path arguments developed in [14].…”

“…To the best of our knowledge, it is still an open problem how to uniquely determine the shape of a bounded perfect conductor with a single incoming wave (for example, a plane wave or an electric dipole). It was proved in [14] that a sound-soft ball can be uniquely determined by an acoustically point source wave. However, it remains unclear to us the unique determination of perfectly conducting ball with an electric dipole.…”

The interior inverse scattering problem for cavities with an artificial obstacle

“…Partial results are known if a priori information on the cavity D is available [17,18]. Recently, Hu and Liu [9] proved that a single point source is sufficient to uniquely determine D provided we known D is a polyhedron or a ball in advance.…”

The reciprocity gap method for a cavity in an inhomogeneous medium

“…In [3], it is proved that the boundary of a homogeneous cavity is uniquely determined from a knowledge of the scattered field u s (x, y) for all x, y ∈ C. The argument can be carried over in a straightforward manner to our case. In some special cases, the shape of the targets can be determined even using a single point source [10]. In this paper, we always assume that k 2 is not a generalized Dirichlet eigenvalue in D c and k is not an exterior transmission eigenvalue.…”