Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

Chandler-Wilde, Simon N.; Heinemeyer, Eric; Potthast, Roland

doi:10.1137/050635262

Cited by 38 publications

(68 citation statements)

References 28 publications

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“…Mathematical theory for scattering from rough unbounded structures was developed from the mid-nineties on by Chandler-Wilde and co-workers starting with theory on integral equations for Dirichlet and impedance rough surface problems for the Helmholtz equation, see, e.g., [5,10,21]. Corresponding results for the same problem in three dimensions are very recent [6][7][8]. Scalar scattering problems involving penetrable media have been considered in [9,11,12,15] both in two and three dimensions.…”

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Electromagnetic Wave Scattering from Rough Penetrable Layers

Haddar¹,

Lechleiter²

2011

SIAM J. Math. Anal.

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Abstract. We consider scattering of time-harmonic electromagnetic waves from an unbounded penetrable dielectric layer mounted on a perfectly conducting infinite plate. This model describes for instance propagation of monochromatic light through dielectric photonic assemblies mounted on a metal plate. We give a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space of functions defined in an unbounded domain containing the dielectric structure. Further, we derive a Rellich identity for a solution to the variational formulation. For simple material configurations and under suitable non-trapping and smoothness conditions, this integral identity allows to prove an a-priori estimate for such a solution. A-priori estimates for solutions to more complicated material configurations are then shown using a perturbation approach. While the estimates derived from the Rellich identity show that the electromagnetic rough surface scattering problem has at most one solution, a limiting absorption argument finally implies existence of a solution to the problem.1. Introduction. Consider an antenna placed over an unbounded penetrable dielectric layer of finite height mounted on a planar metal substrate. Assuming that the antenna operates at fixed frequency, the electromagnetic field caused by the antenna solves a source problem for the time-harmonic Maxwell's equations in the half-space above the substrate. The problem to find this solution to Maxwell's equations when given the source and the dielectric is what we call the electromagnetic rough layer scattering problem. Figure 1.1 illustrates the setting of this problem.

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Electromagnetic Wave Scattering from Rough Penetrable Layers

Haddar¹,

Lechleiter²

2011

SIAM J. Math. Anal.

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“…The extension to the case when ∂D is Lipschitz is outlined in Zhang [28]. To date, however, the only existence result [8] for the three-dimensional rough surface problem, derived via integral equation methods, applies only to the Dirichlet boundary value problem for the Helmholtz equation when the rough surface is the graph of a sufficiently smooth function with sufficiently small surface slope, and deals only with the case when the wave number is sufficiently small.…”

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“…This problem has been studied in a rigorous manner by integral equation methods [10,9,30,3,4,28,8] in the case when Γ is the graph of a sufficiently smooth-bounded function f so that…”

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Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces

Chandler-Wilde¹,

Monk²

2005

SIAM J. Math. Anal.

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128

179

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Abstract. In this paper we study, via variational methods, the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface. The boundary ∂D is assumed to lie within a finite distance of a flat plane and the incident wave is that arising from an inhomogeneous term in the Helmholtz equation whose support lies within some finite distance of the boundary ∂D. Via analysis of an equivalent variational formulation, we provide the first proof of existence of a unique solution to a three-dimensional rough surface scattering problem for an arbitrary wave number. Our method of analysis does not require any smoothness of the boundary which can, for example, be the graph of an arbitrary bounded continuous function. An attractive feature is that all constants in a priori bounds, for example the inf-sup constant of the sesquilinear form, are bounded by explicit functions of the wave number and the maximum surface elevation. 1. Introduction. This paper is concerned with the development and analysis of a variational formulation for scattering by unbounded surfaces, in particular, with the study of what are termed rough surface scattering problems in the engineering literature. We shall use the phrase rough surface to denote surfaces which are a (usually nonlocal) perturbation of an infinite plane surface such that the whole surface lies within a finite distance of the original plane. Such problems arise frequently in applications, for example in modeling acoustic and electromagnetic wave propagation over outdoor ground and sea surfaces, and are the subject of intensive studies in the engineering literature, with a view to developing both rigorous methods of computation and approximate, asymptotic, or statistical methods (see, e.g., the reviews and monographs by Ogilvy In this paper we will focus on a particular, typical problem of the class, which models time harmonic acoustic scattering by a sound soft rough surface. In particular, we seek to solve the Helmholtz equation with wave number k > 0, Δu + k 2 u = g, in the perturbed half-plane or half-space D ⊂ R n , n = 2, 3. We suppose that the homogeneous Dirichlet boundary condition u = 0 holds on ∂D, and a suitable radiation condition is imposed to select a unique solution to this problem. We shall give in the next section complete details about our assumptions on D and on the radiation condition, but we now note that the inhomogeneous term g might be in L 2 (D) with bounded support, or be a more general distribution. In addition the boundary ∂D may or may not be the graph of a function.

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“…For instance, for the case of a bounded obstacle, existence of solution for the time harmonic exterior Dirichlet or impedance scattering problem is known for a long time [13]. For the rough surface scattering problem with a Dirichlet boundary condition, corresponding results have only been achieved during the last decade, firstly by using integral equation approach [4,5,16], and more recently, by using a variational approach in [2,6]. For scattering from rough infinite layers we also refer to recent results in [11].…”

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confidence: 99%

Asymptotic models for scattering from unbounded media with high conductivity

Haddar¹,

Lechleiter²

2010

ESAIM: M2AN

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Abstract.We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of L2 solution operators. We also discuss some numerical experiments concerning these boundary conditions. Mathematics Subject Classification. 35C20, 78A40. IntroductionTime harmonic wave scattering from rough layers is an important problem in science and engineering, as it describes for instance scattering of electromagnetic waves from the ground when one models the earth as a rough stratified medium. In such a model, the moisture of soil causes absorption of the electromagnetic wave inside the ground, and thus naturally leads to a scattering problem for a rough absorbing layer. Since waves inside the absorbing part of the medium decay exponentially with respect to the distance to the layer's boundary, a lot of research has been carried out how to replace the wave scattering problem inside the absorbing layer by some easily handable absorbing boundary condition on the interface in between the absorbing layer and free space [1,8,9,14,15]. The aim of such a boundary condition is to set up an approximate scattering problem merely in the complement of the absorbing object, while still guaranteeing a reliable error bound on the solution of the approximate problem. This error bound depends on what we call the order of the boundary condition as well as on the magnitude of the absorption inside the layer. Indeed, we treat the magnitude of absorption as a parameter and expand the wave field in a power series with respect to the inverse of this parameter. Approximate boundary conditions are built after truncation of this series, the order of the conditions so obtained then corresponds to the truncation index. Truncation at order 0 simply leads to a Dirichlet boundary condition, which is naturally the formal limit condition as the absorption tends to infinity; truncation at order 1 leads to a (usual) impedance Keywords and phrases. Scattering problems, unbounded domains, asymptotic models, generalized impedance boundary conditions, high conductivity. boundary condition. This is the reason why we call the condition arising from truncation at higher order generalized impedance boundary condition (GIBC).In this paper, we analyse GIBCs for rough absorbing layers up to order 3 and shall restrict ourselves to the scalar problem (which corresponds in 2-D to the E-polarization of electromagnetic waves). While the construction of such conditions is rather analogous to the case of a bounded absorbing inhomogeneity, the error analysis is...

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Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

Cited by 38 publications

References 28 publications

Electromagnetic Wave Scattering from Rough Penetrable Layers

Electromagnetic Wave Scattering from Rough Penetrable Layers

Existence, Uniqueness, and Variational Methods for Scattering by Unbounded Rough Surfaces

Asymptotic models for scattering from unbounded media with high conductivity

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