On the diffraction of high-frequency waves by a cone of arbitrary shape

Babich, V. M.; Dement'ev, D. B.; Samokish, B. A.

doi:10.1016/0165-2125(94)00049-b

Cited by 36 publications

(26 citation statements)

References 6 publications

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“…It is indeed only "naturally convergent" for a very restricted range of ordered pairs (ω, ω 0 ). On this restricted range of ordered pairs, (2) can indeed be written as an exponentially convergent integral as shown in [9] and implemented in [10]. In the particular case of an axi-symmetric problem, Babich et al showed in [10] that, using earlier ideas of Nikolaev [25], (2) could be modified into an exponentially convergent integral everywhere.…”

Section: Notations and Problem Formulationmentioning

confidence: 99%

“…On this restricted range of ordered pairs, (2) can indeed be written as an exponentially convergent integral as shown in [9] and implemented in [10]. In the particular case of an axi-symmetric problem, Babich et al showed in [10] that, using earlier ideas of Nikolaev [25], (2) could be modified into an exponentially convergent integral everywhere. In a more recent paper, Shanin [13] improved (2) by introducing three modified Smyshlyaev formulae in the Dirichlet case.…”

Section: Notations and Problem Formulationmentioning

confidence: 99%

“…However, this solution has long been suspected to be wrong (see Meister [7]), and indeed it has recently been disproved by Albani [8]. A different way to attack the quarter-plane problem has been introduced by Smyshlyaev [9], followed by some work by Babich et al [10], but in this case the solution is still difficult to evaluate numerically. Despite this difficulty, Babich et al [11] describe a numerical method based on the Abel-Poissontype summation method and a boundary integral equation that gives the diffraction coefficient for smooth convex cones in the non-singular directions.…”

Section: Introductionmentioning

confidence: 99%

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On the diffraction of acoustic waves by a quarter-plane

Assier

Peake

2012

Wave Motion

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This paper follows the work of A.V. Shanin on diffraction by an ideal quarter-plane. Shanin's theory, based on embedding formulae, the acoustic uniqueness theorem and spherical edge Green's functions, leads to three modified Smyshlyaev formulae, which partially solve the far-field problem of scattering of an incident plane wave by a quarter-plane in the Dirichlet case. In this paper, we present similar formulae in the Neumann case, and describe a numerical method allowing a fast computation of the diffraction coefficient using Shanin's third modified Smyshlyaev formula. The method requires knowledge of the eigenvalues of the Laplace-Beltrami operator on the unit sphere with a cut, and we also describe a way of computing these eigenvalues. Numerical results are given for different directions of incident plane wave in the Dirichlet and the Neumann cases, emphasising the superiority of the third modified Smyshlyaev formula over the other two.

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Section: Notations and Problem Formulationmentioning

confidence: 99%

Section: Notations and Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the diffraction of acoustic waves by a quarter-plane

Assier

Peake

2012

Wave Motion

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“…Then both the scattered wave U sc and the total wave U := U inc + U sc satisfy the three-dimensional Helmholtz equation, (Δ + k 2 )U = 0, in the domain of propagation, and U sc satisfies an appropriate version of the radiation conditions. The theory in [33,4,6] describes the behavior of the diffracted component U diff (x) of U sc (x) at any point x in the domain of propagation. Using spherical coordinates centered at the conical point-x = rω with ω ∈ S 2 and r > 0 denoting the distance of x from the conical point-it follows from the general recipes of the GTD that (with either Dirichlet or Neumann conditions imposed on the surface of the scatterer) U diff has the asymptotic representation…”

Section: Introductionmentioning

confidence: 99%

The Computation of Conical Diffraction Coefficients in High-Frequency Acoustic Wave Scattering

Bonner¹,

Graham²,

Smyshlyaev³

2005

SIAM J. Numer. Anal.

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When a high-frequency acoustic or electromagnetic wave is scattered by a surface with a conical point, the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion, valid as the wave number k → ∞. The diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high-frequency scattering. It can be computed by solving a family of homogeneous boundary value problems for the Laplace-Beltrami-Helmholtz equation (parametrized by a complex wave number-like parameter ν) on a portion of the unit sphere bounded by a simple closed contour , and then integrating the resulting solutions with respect to ν. In this paper we give the numerical analysis of a method for carrying out this computation (in the case of acoustic waves) via the boundary integral method applied on , emphasizing the practically important case when the conical scatterer has lateral edges. The theory depends on an analysis of the integral equation on , which shows its relation to the corresponding integral equation for the planar Helmholtz equation. This allows us to prove optimal convergence for piecewise polynomial collocation methods of arbitrary order. We also discuss efficient quadrature techniques for assembling the boundary element matrices. We illustrate the theory with computations on the classical canonical open problem of a trihedral cone.

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“…However, this solution has long been suspected to be wrong (see Meister, 1987), and indeed it has recently been disproved by Albani (2007). A different way to attack the quarter-plane problem has been introduced by Smyshlyaev (1990), followed by some work by Babich et al (1995), but in this case the solution is still difficult to evaluate numerically. Despite this difficulty, Babich et al (2000) describe a numerical method based on the Abel-Poisson-type summation method and a boundary integral equation that gives the diffraction coefficient for smooth convex cones in the non-singular directions.…”

Section: Introductionmentioning

confidence: 99%

Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane

Assier

Peake

2012

IMA Journal of Applied Mathematics

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This paper provides a review of important results concerning the Geometrical Theory of Diffraction and Geometrical Optics. It also reviews the properties of the existing solution for the problem of diffraction of a time harmonic plane wave by a half-plane. New mathematical expressions are derived for the wave fields involved in the problem of diffraction of a time harmonic plane wave by a quarter-plane, including the secondary radiated waves. This leads to a precise representation of the diffraction coefficient describing the diffraction occurring at the corner of the quarter-plane. Our results for the secondary radiated waves are an important step towards finding a formula giving the corner diffraction coefficient everywhere.

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On the diffraction of high-frequency waves by a cone of arbitrary shape

Cited by 36 publications

References 6 publications

On the diffraction of acoustic waves by a quarter-plane

On the diffraction of acoustic waves by a quarter-plane

The Computation of Conical Diffraction Coefficients in High-Frequency Acoustic Wave Scattering

Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane

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