Time-Harmonic Acoustic Scattering from Locally Perturbed Half-Planes

This paper is concerned with time-harmonic acoustic scattering of plane waves in one or two inhomogeneous half-planes with an unbounded interface. The contrast function is supposed to have a compact support, while the infinite interface is a local perturbation of the x_1-axis. For an acoustically impenetrable interface, the scattering phenomenon occurs in one half-plane only and the impedance (Robin) boundary value problem is investigated. In the penetrable case, we study a transmission problem in two half-planes. Our approach for forward scattering is based on the finite element method in a truncated bounded domain coupled with the boundary element method. Numerical experiments are tested to verify our scheme. For the inverse problem, we prove that the near-field data of a finite number of incoming plane waves or a single point source wave uniquely determine the shape of a rectangular cavity of impedance-type

A highly accurate perfectly-matched-layer boundary integral equation solver for acoustic layered-medium problems

Based on the perfectly matched layer (PML) technique, this paper develops a high-accuracy boundary integral equation (BIE) solver for acoustic scattering problems in locally defected layered media in both two and three dimensions. The original scattering problem is truncated onto a bounded domain by the PML. Assuming the vanishing of the scattered field on the PML boundary, we derive BIEs on local defects only in terms of using PML-transformed free-space Green's function, and the four standard integral operators: single-layer, double-layer, transpose of double-layer, and hyper-singular boundary integral operators. The hyper-singular integral operator is transformed into a combination of weakly-singular integral operators and tangential derivatives. We develop a high-order Chebyshev-based rectangular-polar singular-integration solver to discretize all weakly-singular integrals. Numerical experiments for both two- and three-dimensional problems are carried out to demonstrate the accuracy and efficiency of the proposed solver.Comment: 19 pages, 16 figure

Acoustic scattering from locally perturbed periodic surfaces

We prove well-posedness for the time-harmonic acoustic scattering of plane waves from locally perturbed periodic surfaces in two dimensions under homogeneous Dirichlet boundary conditions. This covers sound-soft acoustic as well as perfectly conducting, TE polarized electromagnetic boundary value problems. Our arguments are based on a variational method in a truncated bounded domain coupled with a boundary integral representation. If the quasi-periodic Green's function to the unperturbed periodic scattering problem is calculated efficiently, then the variational approach can be used for a numerical scheme based on coupling finite elements with a boundary element algorithm. Even for a general 2D rough-surface problem, it turns out that the Green's function defined with the radiation condition ASR satisfies the Sommerfeld radiation condition over the half plane. Based on this result, for a local perturbation of a periodic surface, the scattered wave of an incoming plane wave is the sum of the scattered wave for the unperturbed periodic surface plus an additional scattered wave satisfying Sommerfeld's condition on the half plane. Whereas the scattered wave for the unperturbed periodic surface has a far field consisting of a finite number of propagating plane waves, the additional field contributes to the far field by a far-field pattern defined in the half-plane directions similarly to the pattern known for bounded obstacles