Complete families in a given function space are sets of linearly independent functions, a linear combination of which can approximate any other function with arbitrarily high accuracy. Outgoing cylindrical wave functions are one such family, used to represent the scattered wave in exterior boundary value problems for the scalar Helmholtz equation in two spatial dimensions. When the incident wave is plane and the scattered wave is represented by a series of said functions, which converges up to the boundary of the obstacle, the obstacle is said to be in the Rayleigh class. One shall further distinguish between Dirichlet-Rayleigh and Neumann-Rayleigh obstacles, according to the applicable boundary condition. Discs are trivial obstacles of these classes. Ellipses of eccentricity n such that n2 < 1/2 were shown to be in the Dirichlet-Rayleigh class by Barantsev et al. in 1971, who used the saddle point method to asymptotically estimate the Fourier scattering coefficients. Herewith, another one parameter family of obstacles is constructed by the same method. It is also shown that the same obstacles are in the Neumann-Rayleigh class. The relevance of these results to the numerical treatment of scattering problems is briefly discussed.

Crosta, G. (2001). Complete families and Rayleigh obstacles. JOURNAL OF COMPUTATIONAL ACOUSTICS, 9(2), 611-622 [10.1142/S0218396X01000589].

Complete families and Rayleigh obstacles

CROSTA, GIOVANNI FRANCO FILIPPO
Primo
2001

Abstract

Complete families in a given function space are sets of linearly independent functions, a linear combination of which can approximate any other function with arbitrarily high accuracy. Outgoing cylindrical wave functions are one such family, used to represent the scattered wave in exterior boundary value problems for the scalar Helmholtz equation in two spatial dimensions. When the incident wave is plane and the scattered wave is represented by a series of said functions, which converges up to the boundary of the obstacle, the obstacle is said to be in the Rayleigh class. One shall further distinguish between Dirichlet-Rayleigh and Neumann-Rayleigh obstacles, according to the applicable boundary condition. Discs are trivial obstacles of these classes. Ellipses of eccentricity n such that n2 < 1/2 were shown to be in the Dirichlet-Rayleigh class by Barantsev et al. in 1971, who used the saddle point method to asymptotically estimate the Fourier scattering coefficients. Herewith, another one parameter family of obstacles is constructed by the same method. It is also shown that the same obstacles are in the Neumann-Rayleigh class. The relevance of these results to the numerical treatment of scattering problems is briefly discussed.
Articolo in rivista - Articolo scientifico
Applied Mathematics; Acoustics and Ultrasonics; scattering; complex analysis; integral equation; stationary phase method; asymptotic analysis
English
611
622
12
Thanks to Prof. T. Wirgin for suggesting the study of the Neumann obstacle; Mathematical Review MR1853645 Crosta, Giovanni F. Complete families and Rayleigh obstacles. J. Comput. Acoust. 9 (2001), no. 2, 611–622. (Reviewer: B. Belinskiy) ``The series of outgoing cylindrical wave functions play an important role in the solution of many direct and inverse problems of the scattering theory in the 2D case. The well-known Rayleigh hypothesis claims that the solution of the scattering problem on an obstacle may be represented as such a series that converges up to the boundary of the obstacle. It appears that the hypothesis is correct only for some classes of obstacles. The description of the obstacles that are in the Rayleigh class and the corresponding families of outgoing wave functions is therefore of great interest. The paper contains a description of a new class of those obstacles. The problem of scattering is reduced to an integral equation. The polar angle in that equation is considered in the complex plane and the singularities of the solution are analyzed there. The method of steepest descents allows finding the asymptotic representation for the Fourier coefficients of the series under consideration and hence proving convergence for some obstacles.''
Crosta, G. (2001). Complete families and Rayleigh obstacles. JOURNAL OF COMPUTATIONAL ACOUSTICS, 9(2), 611-622 [10.1142/S0218396X01000589].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/93254
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