Elastic scattering from rough surfaces in three dimensions

Hu, Guanghui; Li, Peijun; Zhao, Yue

doi:10.1016/j.jde.2020.03.022

Cited by 4 publications

(3 citation statements)

References 37 publications

(82 reference statements)

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“…The problem in three dimensions is similar but much more complicated than that in two dimensions. Based on uniqueness of the elastic scattering by deterministic rough surface in three dimensions proved by Hu et al [18] and corresponding upward radiation condition and Dirichlet to Neumann operator defined for three-dimensional case, the results in this paper can be similarly but more complicatedly extended to three dimensions together with some prerequisites such as Lemmas 3.1 and 3.2, the Rellich identities and the a priori bound for the Helmholtz equation in [10] for three-dimensional case. Future work will focus on elastic scattering with an incident plane wave, which is still remained unsolved since the Rellich identity is not valid any more and additional difficulties arise in this case.…”

Section: Discussionmentioning

confidence: 90%

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A Priori Bounds for Elastic Scattering by Deterministic and Random Unbounded Rough Surfaces

Wang¹,

Xu²

2023

CSIAM-AM

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This paper investigates the elastic scattering by unbounded deterministic and random rough surfaces, both of which are assumed to be graphs of Lipschitz continuous functions. For the deterministic case, an a priori bound explicitly dependent on frequencies is derived by the variational approach. For the scattering by random rough surfaces with a random source, well-posedness of the corresponding variation problem is proved. Moreover, a similar bound with explicit dependence on frequencies for the random case is also established based upon the deterministic result, Pettis measurability theorem and Bochner's integrability theorem.

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Section: Discussionmentioning

confidence: 90%

“…Furthermore, they studied solvability in weighted Sobolev spaces, on which they based to prove the existence and uniqueness of elastic scattering by unbounded rough surfaces with a plane or point source incident wave in [15]. Recently Hu et al [18] generalized the similar results for three-dimensions.…”

Section: Introductionmentioning

confidence: 99%

A Priori Bounds for Elastic Scattering by Deterministic and Random Unbounded Rough Surfaces

Wang¹,

Xu²

2023

CSIAM-AM

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“…The paper [23] extends the method in [4] to the scattering by finite height inhomogeneous layer with the Nuemann and generarized impedance boundary boundary. The papers [18,19,21,26,22] also consider the scattering by electromagnetic and elastic waves.…”

Section: Introductionmentioning

confidence: 99%

A Nyström method for scattering by a two-layered medium with a rough boundary

Liu¹,

Li²,

Yang³

et al. 2023

Preprint

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This paper presents a study on the integral equation method and the Nystöm method for the scattering of time-harmonic acoustic waves by a two-layered medium with an unbounded perturbed boundary. The medium consists of two layers separated by a plane interface, for which we assume transmission boundary conditions. We assume either Dirichlet or impedance boundary conditions for the rough surface boundary. Unlike classical rough surface scattering problems, the presence of a plane interface makes it difficult to establish the well-posedness of the scattering problem and to find a numerical treatment. We introduce the two-layered Green function and prove that this function has similar asymptotic decay properties to the half-space Green function. By using a similar approach to classical rough surface problems, we establish the uniqueness of the scattering problem. We derive the integral equation formulations using the two-layered Green function as the integral kernel and use them to prove the existence of the scattering problem. Furthermore, we propose the Nystöm method for discretizing the integral equations and establish its convergence. Finally, we perform numerical experiments to demonstrate the effectiveness of the Nystöm method.

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