Time-Harmonic Acoustic Scattering from a Nonlocally Perturbed Trapezoidal Surface

Abstract

This paper is concerned with acoustic scattering from a sound-soft trapezoidal surface in two dimensions. The trapezoidal surface is supposed to consist of two horizontal halflines pointing oppositely, and a single finite vertical line segment connecting their endpoints, which can be regarded as a non-local perturbation of a straight line. For incident plane waves, we enforce that the scattered wave, post-subtracting reflected plane waves by the two half lines of the scattering surface in certain two regions respectively, satisfies an integral form of Sommerfeld radiation condition at infinity. With this new radiation condition, we prove uniqueness and existence of weak solutions by a coupling scheme between finite element and integral equation methods. This consequently indicates that our new radiation condition is sharper than the Angular Spectrum Representation, and has generalized the radiation condition for scattering problems in a locally perturbed half-plane.Furthermore, we develop a numerical mode matching method based on this new radiation condition. A perfectly matched layer is setup to absorb outgoing waves at infinity. Since the medium composes of two horizontally uniform regions, we expand, in either uniform region, the scattered wave in terms of eigenmodes and match the mode expansions on the common interface between the two uniform regions, which in turn gives rise to numerical solutions to our problem. Numerical experiments are carried out to validate the new radiation condition and to show the performance of our numerical method.

Mathematical analysis of wave radiation by a step-like surface

“…The theory reveals that the scattered wave, post-subtracting a known wave field, satisfies the same uSRC, so that its far-field pattern is accessible theoretically. For a plane-wave incidence, asymptotic analysis shows that merely subtracting reflected plane waves, due to nonuniform heights of the step-like surface at infinity, from the scattered wave in respectively affected regions, as was done in the previous paper [20], produces a radiating wave, though discontinuous, satisfies the uSRC as well. Numerically, we adopt a previously developed PML-based BIE method [21] to solve the problem.…”

“…For a cylindrical-wave incidence due to a line source, our previous work [16] has shown that for a general rough surface, the scattered wave u sc directly satisfies the integral form of SRC, a weakened SRC (wSRC); we also show that if a horizontal stripe is removed from the scattering domain, then u sc satisfies the classic SRC, a pointwise but stronger condition, in the remaining half plane. But for a plane-wave incidence, no SRC condition has been imposed for wave scattered by the unbounded curve ∂Ω, until recently the author and Hu in [20] proved that the scattered wave u sc piecewisely satisfies wSRC at infinity. Due to the absence of a background Green function, the wSRC condition couldn't strictly characterize the radiation behavior of u sc at infinity, and hence the far field pattern of u sc is unclear; moreover, we couldn't explain why the PML truncation used there should produce a physically correct solution.…”

“…We point out that Ω h is not a suitable background since the vertical segment of ∂Ω h makes Fourier transforming x 1 -variable impossible. We rigorously analyze asymptotic behavior and far-field pattern of the Green function G at infinity, showing that it satisfies a universal-direction SRC (uSRC) stronger than that in [20,16] in the sense that G satisfies the classic SRC condition uniformly along all directions in the full region Ω b , with no restrictions on the satisfied region. Based on this, we propose, for either a cylindrical incident wave due to a line source or a plane incident wave, an implicit TBC (ITBC) to terminate the unbounded domain Ω such that a wellposedness theory is established via an associated variational formulation.…”

On Wellposedness and Convergence of UPML method for analyzing wave scattering in layered media

“…One issue is that the real path used in (27)(28)(29)(30) is not usable to perform the extension since e i(x 1 −ỹ 1 )ξ blows up in one of the two cases ξ → ±∞. To resolve this, we make use of Lemma 3.7 by changing the real path to EXT : +∞i → 0 → +∞, so that we can define, for instance,…”

“…Examples include optical waveguides, near field imaging, communication with submarine, detection of buried objects and so on. As a result, the analysis and numerical computation of layered medium scattering problems have been constantly attracting attentions from researchers both in engineering and mathematical communities [4,18,22,35,3,29,30,41].…”