Journal of Mathematical Sciences

, Volume 214, Issue 3, pp 322–336 | Cite as

Integral Equations and the Scattering Diagram in the Problem of Diffraction by Two Shifted Contacting Wedges with Polygonal Boundary

  • M. A. Lyalinov

The acoustic problem of diffraction by two wedges with different wave velocities is studied. It is assumed that the wedges with parallel edges have a common part of the boundary and the second wedge is shifted with respect of the first one in the orthogonal to the edges direction along the common part of the boundary. The wave field is governed by the Helmholtz equations. On the polygonal boundary, separating these shifted wedges from the exterior, the Dirichlet boundary condition is satisfied. The wave field is excited by an infinite filamentary source, which is parallel to the edges. In these conditions, the problem is effectively two-dimensional. The Kontorovich–Lebedev transform is applied to separate the radial and angular variables and to reduce the problem at hand to integral equations of the second kind for so-called spectral functions. The kernel of the integral equations given in the form of an integral of the product of Macdonald functions is analytically transformed to a simplified expression. For the problem at hand, some reductions of the equations are also discussed for the limiting or degenerate values of parameters. Making use of an alternative integral representation of the Sommerfeld type, expressions for the scattering diagram are then given in terms of spectral functions. Bibliography: 24 titles.


Spectral Function Singular Integral Equation Cylindrical Wave Fredholm Property Polygonal Boundary 
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  1. 1.
    A. K. Gautesen, “Scattering of a Rayleigh wave by an elastic quarter space-revisited,” Wave Motion, 35, 91–98 (2002a).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. K. Gautesen, “Scattering of a Rayleigh wave by an elastic quarter space-revisited,” Wave Motion, 35, 417–424 (2002b).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    K. Fujii, “Rayleigh-wave scattering of various wedge corners: Investigation in the wider range of wedge angles,” Bull. Seismol. Soc. Am., 84, 1916–1924 (1994).Google Scholar
  4. 4.
    J.-P. Croisille and G. Lebeau, “Diffraction by an immersed elastic wedge,” Lect. Notes Math., 1723, Springer-Verlag, Berlin (1999).Google Scholar
  5. 5.
    V. V. Kamotski and G. Lebeau, “Diffraction by an elastic wedge with stress free boundary: Existence and uniqueness,” Proc. Roy. Soc. A, 462 (2065), 289–317 (2006).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. A, 455, 2655–2686 (1999).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proc. Roy. Soc. A, 462 (2072), 2503–2522 (2006).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    D. S. Jones, “Rawlin’s method and the diaphanous cone,” Quaterly J. Mech. Appl. Math., 53, No. 1, 91–109 (2000).CrossRefMATHGoogle Scholar
  9. 9.
    M. A. Lyalinov, “Acoustic scattering of a plane wave by a circular penetrable cone,” Wave Motion, 48, 1, 62–82 (doi:10.1016/j. wavemoti. 2010.07.002) (2011).Google Scholar
  10. 10.
    J.-M. L. Bernard, “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 (1997).Google Scholar
  11. 11.
    M. A. Lyalinov and N. Y. Zhu, Scattering of Waves by Wedges and Cones with Impedance Boundary Conditions, SciTech-IET Edison, NJ (2012).Google Scholar
  12. 12.
    12. J.-M. L. Bernard and M. A. Lyalinov, “Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section ”, Wave Motion, 33, 155–181 (2001). (Erratum : p.177 replace O(1/cos(π(v − b))) by O(v d sin(πv)/cos(π(v − b))).Google Scholar
  13. 13.
    J.-M. L. Bernard and M. A. Lyalinov, “Electromagnetic scattering by a smooth convex impedance cone,” IMA J. Appl. Math., 69(3), 285–333, June (2004). (Multiply sin(ζ) by n/|n| in (D.20) of Appendix D.)Google Scholar
  14. 14.
    B. V. Budaev, “Diffraction by wedges,” Pitman Research Notes Math., 322, Longman Scientific and Technical, Essex (1995).Google Scholar
  15. 15.
    V. M. Babich, M. A. Lyalinov, and V. E. Grikurov, Diffraction Theory. The Sommerfeld-Malyuzhinets Technique, Alpha Science, Oxford (2008).Google Scholar
  16. 16.
    Y. A. Kravtsov and N. Y. Zhu, Theory of Diffraction. Heuristic Approach, Alpha Science, Oxford (2010).Google Scholar
  17. 17.
    J.-M. L. Bernard, “A spectral approach for scattering by impedance polygons,” Q. Jl. Mech. Appl. Math., 59(4), 517–550 (2006).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    A. S. Fokas, “Two Dimensional Linear PDEs in a Convex Polygon,” Proc. R. Soc. Lond. A, 457, 371–393 (2001).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    B. V. Budaev and D. B. Bogy, “Diffraction by a convex polygon with side-wise constant impedance,” Wave Motion, 43 (8), 631–645 (2006).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    D. S. Jones, “The Kontorovich-Lebedev transform,” J. Inst. Maths Applics, 26, 133–141 (1980).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    A. D. Avdeev and S. M. Grudsky, “On modified Kontorovich-Lebedev transform and its application to the problem of cylindrical wave diffraction on perfectly conducting wedge,” Radiotekhnika Elektronika, 39(7), 1081–1089 (1994).Google Scholar
  22. 22.
    I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., Academic Press, Orlando (1980).Google Scholar
  23. 23.
    L. S. Rakovschik, “Systems of integral equations with almost difference operators,” Sibirian Math. J., 3, No. 2, 250–255 (1962).MathSciNetGoogle Scholar
  24. 24.
    V. M. Babich, D. B. Dement’ev, B. A. Samokish, and V. P. Smyshlyaev, “On evaluation of the diffraction coefficients for arbitrary ‘nonsingular’ directions of a smooth convex cone,” SIAM J. Appl. Math., 60(2), 536–573 (2000).MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of St.Petersburg State UniversitySt.PetersburgRussia

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