Acoustic scattering by a circular semi-transparent conical surface. (English) Zbl 1131.76047

Summary: We study the scattering of a plane acoustic wave by a circular semi-transparent conical surface with impedance-type boundary conditions. An analytic solution is constructed on the basis of incomplete separation of variables and the reduction of the problem to a functional-difference equation of the second order. Although the latter is equivalent to a Carleman boundary value problem for analytic vectors, the solution is studied by means of the direct reduction method, that is, converting the functional-difference equations to a Fredholm-type integral equation. Its unique solvability is then studied, and an expression for the scattering amplitude of spherical wave from the vertex is discussed. Some numerical results are also presented for axial incidence.


76Q05 Hydro- and aero-acoustics
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[1] Felsen LB (1957). Plane wave scattering by small-angle cones. IRE Trans Antennas Propagation 5: 121–129
[2] Borovikov VA (1966). Diffraction by polygons and polyhedrons. Nauka, Moscow
[3] Cheeger J, Taylor ME (1982) Diffraction of waves by conical singularities. Commun Pure Appl Math 35(3,4): 275–331, 487–529 · Zbl 0526.58049
[4] Smyshlyaev VP (1990). Diffraction by conical surfaces at high frequencies. Wave Motion 12(4): 329–339 · Zbl 0721.73011
[5] Jones DS (1997). Scattering by a cone. Quart J Mech Appl Math 50: 499–523 · Zbl 0958.78014
[6] Babich VM, Dement’ev DB, Samokish BA and Smyshlyaev VP (2000). On evaluation of the diffraction coefficients for arbitrary ’non-singular’ directions of a smooth convex cone. SIAM Appl Math 66(2): 536–573
[7] Bernard J-ML (1997) Méthode analytique et transformées fonctionnelles pour la diffraction d’ondes par une singularité conique: équation intégrale de noyau non oscillant pour le cas d’impédance constante. rapport CEA-R-5764, Editions Dist-Saclay
[8] Bernard J-ML, Lyalinov MA (2001) Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section. Wave Motion 33:155–181 (erratum: p 177 replace \(O(1/\cos(\pi(\nu-b)))\) by \(O(\nu^d\sin(\pi\nu)/\cos(\pi(\nu-b)))\) ) · Zbl 1074.78516
[9] Antipov YA (2002). Diffraction of a plane wave by a circular cone with an impedance boundary condition. SIAM J Appl Math 62(4): 1122–1152 · Zbl 1004.65121
[10] Lyalinov MA (2003) Diffraction of a plane wave by an impedance cone. Zap Nauch Sem Peterburg Otd Matem Inst Steklova) 297 Matem vopr teorii rasprostr voln 32:191–215 [English transl in: J Math Sci 2005, 127(6):2446–2460]
[11] Bernard J-ML and Lyalinov MA (2004). Electromagnetic scattering by a smooth convex impedance cone. IMA J Appl Math 69: 285–333 · Zbl 1115.78005
[12] Daniele VG (2003). Winer–Hopf technique for impenetrable wedges having arbitrary aperture angle. SIAM J Appl Math 63(4): 1442–1460 · Zbl 1032.78008
[13] Abrahams ID and Lawrie JB (1995). On the factorization of a class of Winer–Hopf kernels. IMA J Appl Math 55(1): 35–47 · Zbl 0841.35024
[14] Antipov YA and Silvestrov VV (2004). Second-order functional-difference equations I: method of the Riemann–Hilbert problem on Riemann surfaces. Quart J Mech Appl Math 57(2): 245–265 · Zbl 1064.39016
[15] Lyalinov MA and Zhu NY (2003). A solution procedure for second-order difference equations and its application to electromagnetic-wave diffraction in a wedge-shaped region. Proc Roy Soc Lond A 459: 3159–3180 · Zbl 1092.78008
[16] Lyalinov MA and Zhu NY (2006). Diffraction of a skew incident plane electromagnetic wave by an impedance wedge. Wave Motion 44: 21–43 · Zbl 1231.78028
[17] Buslaev VS and Fedotov AA (2001). On the difference equations with periodic coefficients. Adv Theor Math Phys 5: 1105–1168 · Zbl 1012.39008
[18] Senior TBA and Volakis JL (1995). Approximate boundary conditions in electromagnetics. Inst of Electr Eng, London · Zbl 0828.73001
[19] Buldyrev VS and Lyalinov MA (2001). Mathematical methods in modern electromagnetic diffraction theory, vol 1. Intern monographs on advanced electromagnetics Science House, Tokyo
[20] Abramowitz M, Stegun I (1972) Handbook of mathematical functions. Dover Publications · Zbl 0543.33001
[21] Gradshteyn IS and Ryzhik IM (1980). Table of integrals, series and products, 4th edn. Academic Press, Orlando · Zbl 0521.33001
[22] Jones DS (1964). The theory of electromagnetism. Pergamon Press, London · Zbl 0121.21604
[23] Tuzhilin AA (1973). The theory of Maliuzhinets’ inhomogeneous functional equations. Different Uravn 9(10): 1875–1888
[24] Jones DS (1980). The Kontorovich–Lebedev transform. J Inst Math Appl 26: 133–141 · Zbl 0449.44003
[25] Titchmarsh EC (1937) Introduction to the theory of Fourier integrals. Oxford Press · Zbl 0017.40404
[26] Babich VM (2002). To the problem on the asymptotics with respect to indices of the associated Legendre functions. Russian J Math Phys 9(1): 6–13, (in English) · Zbl 1104.33300
[27] Vekua NP (1970). Systems of singular integral equations. Nauka, Moscow, (1st edn. 1967 Noordhoff Groningen)
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