A Survey of Open Cavity Scattering Problems

Abstract

This paper gives a brief survey of recent developments on mathematical modeling and analysis of the open cavity scattering problems, which arise in diverse scientific areas and have significant industrial and military applications. The scattering problems are studied for the two-dimensional Helmholtz equation corresponding to the transverse magnetic or electric polarization, and the three-dimensional time-harmonic and time-domain Maxwell equations. Since these problems are imposed in open domains, a key step of the analysis is to develop transparent boundary conditions and reformulate them equivalently into boundary value problems in bounded domains. The well-posedness of weak solutions are shown for the associated variational problems by using either the Lax-Milgram theorem or the Fredholm alternative. Mathematics subject classification: 35Q61, 78A25, 78M30

Time-Harmonic Acoustic Scattering from Locally Perturbed Half-Planes

“…If Γ coincides with x 2 = 0, it holds that v sc = 0 and thus v = g ± . In the case of local perturbations, the existence and uniqueness of v sc follow from existing arguments for cavity scattering problems; see [1,33,27]. Note that the constructed solution v satisfies the UASR but not the Sommerfeld radiation condition.…”

“…In a vast literature (see, e.g., [33,1,27]), it has been proved that the perturbed acoustic scattering problems (see section 2.2 below for the formulation) with the Dirichlet and Neumann boundary conditions admit a unique solution u ∈ H 1 loc (Ω + ) of the form…”

“…In the time-harmonic regime, it is well known that the total field can be decomposed into three parts: the incoming wave u in , the reflected wave u re corresponding to the unperturbed scattering interface (i.e., the straight line x 2 = 0 in 2D), and the scattered wave u sc caused by the presence of local perturbations. Under the Sommerfeld radiation condition of u sc in the half-space, one can show uniqueness and the existence of weak solutions if the total field fulfills a Dirichlet or Neumann boundary condition, whereas the reflected waves are usually uniquely determined by Snell's law in physics; see, e.g., [1,2,3,33,24,26,27,32,36].…”

“…Due to the local perturbation of the infinite interface, variational and integral equation methods (see, e.g., [1,2,27,36,33]) can be adopted to reduce the unbounded physical domain to a truncated computational domain. This has significantly simplified the calculation and analysis over an unbounded domain for treating general rough surface scattering problems [11,7,9,20,22,35,32].…”

“…In fact, a transparent boundary condition (or nonreflecting boundary operator) always requires a series expansion of half-plane Sommerfeld radiation solutions with the corresponding boundary condition on x 2 = 0. However, unlike the Dirichlet and Neumann cases [27,33], it is not trivial to derive such an expansion for the Robin boundary problem. On the other hand, to the best of our knowledge, it remains unclear how to apply the integral equation approach, with the free-space fundamental solution being the kernel (see [36] in the Dirichlet case), to a bounded computational domain for treating the Robin boundary value problem in a locally perturbed half-space.…”