Scattering by Rough Surfaces

DeSanto, John A.

doi:10.1016/b978-012613760-6/50003-6

Cited by 24 publications

(34 citation statements)

References 171 publications

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“…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 [23], Voronovich [26], Saillard and Sentenac [24], Warnick and Chew [27], and de Santo [13]). …”

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confidence: 99%

“…A requirement that (2.9) holds is commonly used (e.g., [13]) as a formal radiation condition in the physics and engineering literature on rough surface scattering. The meaning of (2.9) is clear when…”

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confidence: 99%

“…We also establish the equivalent variational formulation that we use and study in this paper. As part of the boundary value problem formulation we require the radiation condition often used in a formal manner in the rough surface scattering literature (e.g., [13]) that, above the rough surface and the support of g, the solution can be represented in integral form as a superposition of upward traveling and evanescent plane waves. This radiation condition is equivalent to the upward propagating radiation condition proposed for two-dimensional (2D) rough surface scattering problems in [11], and has recently been analyzed carefully in the 2D case by Arens and Hohage [5].…”

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

Chandler-Wilde¹,

Monk²

2005

SIAM J. Math. Anal.

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

Chandler-Wilde¹,

Monk²

2005

SIAM J. Math. Anal.

128

179

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“…Although the equations look like the standard results [3], they differ due to the Born term w 1 and the image Green's function G…”

Section: Continuity Conditions and Coupled Integral Equationsmentioning

confidence: 98%

“…For q ¼ 0 (Dirichlet problem) (24) is an identity. For other review papers and books on these topics see [3,[7][8][9][10].…”

Section: Continuity Conditions and Coupled Integral Equationsmentioning

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Exact boundary integral equations for scattering of scalar waves from infinite rough interfaces

DeSanto

2010

Wave Motion

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Analysis of the scattering by an unbounded rough surface

Shen

2012

Math Methods in App Sciences

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Communicated by A. KirschThis paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two-point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach.

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Scattering by Rough Surfaces

Cited by 24 publications

References 171 publications

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

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

Exact boundary integral equations for scattering of scalar waves from infinite rough interfaces

Analysis of the scattering by an unbounded rough surface

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