A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem

Abstract

. Introduction. A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space U 3 . We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

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

“…This property leads to uniqueness in determining sound-soft balls with an incoming point source wave. Our mathematical analysis is based on the Schwarz reflection principle for harmonic functions [12,19] combined with the constructive method for solving the exterior Dirichlet boundary value problem for the Helmholtz equation for balls in [4,6].…”

“…A novelty in the proof of lemma 2.2 is the derivation of the singularity of the harmonic extension ṽ based on the singularity of v. To prove lemma 2.1, we shall follow the spirit of Colton [6] by constructing solutions to the Helmholtz equation in terms of harmonic functions. The singularity of u at y has to be appropriately treated.…”

“…In [6], the problem of (2.12) and (2.14) was transformed into a Goursat problem for a hyperbolic equation in a cone. Consequently, the well-posedness of analytic solutions of K r s ( , ) in < >…”

Uniqueness of scatterer in inverse acoustic obstacle scattering with a single incident plane wave

“…There is already a vast literature on inverse acoustic scattering problems using the scattering amplitude A(β, α, k), and many progresses have been made in this problem. We refer the readers to [7], [8], [21], [34], [6], [10], [39], [23], [26], [4], [1], [29], [28] [14], [5], [38], [9] [17], [18], [2], [12], [3] and [19] and the references therein for the studies on uniqueness of this inverse problem.…”