Explicit Solution Formulas for the Acoustic Diffraction Problem with a Slit in a Hard and a Soft Screen

  • Matthias Kunik
Chapter

Abstract

We consider the boundary operators for acoustic slit diffraction problems with a hard and soft screen, respectively. This model can also be applied to Sommerfeld’s boundary operators with a Hankel kernel for the diffraction of light through a slit. For a logarithmic approximation of the Hankel kernel we use the Fourier method to derive explicit solutions together with certain regularity results.

Keywords

Explicit Solution Boundary Integral Equation Continuous Extension Diffraction Problem Boundary Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of mathematical functions. Dover Publications, Inc., Mineola (1968)Google Scholar
  2. 2.
    Bastos, M.A., dos Santos, A.F.: Convolution equations of the first kind on a finite interval in Sobolev spaces. Integral Equations and Operator Theory 13, 638–659 (1990)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Belward, J.A.: The solution of an integral equation of the first kind on a finite interval. Quart. Appl. Math. 27, 313–321 (1969)MATHMathSciNetGoogle Scholar
  4. 4.
    Born, M., Wolf, E.: Principles of Optics: Electromagnetic Theory of Propagation, 7th edn. Interference and Diffraction of Light. Cambridge University Press, Cambridge (1999)Google Scholar
  5. 5.
    Castro, L., Kapanadze, D.: Pseudo-differential Operators in a Wave Diffraction Problem with Impedance Conditions. Fractional Calculus & Applied Analysis 11(1), 15–26 (2008)MATHMathSciNetGoogle Scholar
  6. 6.
    Castro, L., Kapanadze, D.: The Impedance Boundary-Value Problem of Diffraction by a Strip. J. Math. Anal. Appl. 337, 1031–1040 (2008)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Castro, L., Kapanadze, D.: Dirichlet-Neumann-impedance boundary-value problems arising in rectangular wedge diffraction problems. Proceedings of the American Mathematical Society 136, 2113–2123 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dörr, J.: Zwei Integralgleichungen erster Art, die sich mit Hilfe Mathieuscher Funktionen lösen lassen. ZAMP III, 427–439 (1952)CrossRefGoogle Scholar
  9. 9.
    Gorenflo, N.: A new explicit solution method for the diffraction through a slit - Part 2. ZAMP 58, 16–36 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gorenflo, N.: A new explicit solution method for the diffraction through a slit. ZAMP 53, 877–886 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gorenflo, N.: A characterization of the range of a finite convolution operator with a Hankel kernel. Integral Transforms and Special Functions 12, 27–36 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gorenflo, N., Kunik, M.: A new and self-contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations, Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg (preprint, 4/2009)Google Scholar
  13. 13.
    Gorenflo, N., Werner, M.: Solution of a finite convolution equation with a Hankel kernel by matrix factorization. SIAM J. Math. Anal. 28, 434–451 (1997)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, Corrected and Enlarged Edition. Academic Press, New York (1980)Google Scholar
  15. 15.
    Kunik, M., Skrzypacz, P.: Diffraction of light revisited. Math. Methods in the Appl. Sci. 31(7), 793–820 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Latta, G.E.: The solution of a class of integral equations. J. Rational Mech. Anal. 5, 821–834 (1956)MathSciNetGoogle Scholar
  17. 17.
    Meister, E.: Einige gelöste und ungelöste kanonische Probleme der mathematischen Beugungstheorie. Expositiones Mathematicae 5, 193–237 (1987)MATHMathSciNetGoogle Scholar
  18. 18.
    Meister, E., Santos, P.A., Teixeira, F.S.: A Sommerfeld-type diffraction problem with second-order boundary conditions. Z. Angew. Math. Mech. 72(12), 621–630 (1992)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Noble, B.: Methods based on the Wiener-Hopf technique for the solution of partial differential equations. Pergamon Press, New York (1958)MATHGoogle Scholar
  20. 20.
    Pal’cev, B.V.: A generalization of the Wiener-Hopf method for convolution equations on a finite interval with symbols having power-like asymptotics at infinity. Math. USSR Sbornik 41(3), 289–328 (1982)CrossRefGoogle Scholar
  21. 21.
    dos Santos, A.F., Teixeira, F.S.: The Sommerfeld problem revisited: Solution spaces and the edge conditionsGoogle Scholar
  22. 22.
    Moura Santos, A., Speck, F.-O.: Sommerfeld diffraction problems with oblique derivatives. Math. Methods in the Appl. Sci. 20(7), 635–652 (1997)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Schmeidler, W.: Integralgleichungen mit Anwendungen in Physik und Technik, Band I, Lineare Integralgleichungen, Akademische Verlagsgesellschaft, Geest & Portig K.-G., Leipzig (1950)Google Scholar
  24. 24.
    Shanin, A.V.: Three theorems concerning diffraction by a strip or a slit. Q. J. Mech. Appl. Math. 54, 107–137 (2001)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Shanin, A.V.: To the problem of diffraction on a slit: Some properties of Schwarzschild’s series. In: Proceedings of the International Seminar Day on Diffraction Millennium Workshop, St. Petersburg, Russia, pp. 143–155 (2000)Google Scholar
  26. 26.
    Sommerfeld, A.: Mathematische Theorie der Diffraction. Math. Ann. 47(2,3), 317–374 (1896)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sommerfeld, A.: Vorlesungen über Theoretische Physik, Band IV, Optik. Verlag Harri Deutsch (1989)Google Scholar
  28. 28.
    Spahn, R.: The diffraction of a plane wave by an infinite slit. I. Quart. Appl. Math. 40(1), 105–110 (1982/83)MathSciNetGoogle Scholar
  29. 29.
    Williams, M.H.: Diffraction by a finite strip. Q. J. Mech. Appl. Math. 35(1), 103–124 (1982)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Matthias Kunik
    • 1
  1. 1.Institut für Analysis und NumerikOtto-von-Guericke-UniversitätMagdeburgGermany

Personalised recommendations