A Far-Field Asymptotic Analysis in the High-Frequency Diffraction by Cracks

  • M. Y. RemizovEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 109)


On the basis of recently obtained asymptotic solutions of integral equations by the Wiener-Hopf method for diffraction by a straight finite-length crack in a linear elastic medium, we study the properties of the far-zone scattered field at high frequencies for (1) - anti-plane problem in a homogeneous medium, (2) - anti-plane problem for an interface crack, and (3) - in-plane problem in a homogeneous medium.The method proposed is founded on a high-frequency solution of the basic integral equation of the scattering problem. Then we develop an explicit analytical representation for the leading asymptotic term, by estimating the far-field behavior of the relevant integrals with high oscillations by the method of stationary phase. This allows us to obtain the final form of the scattered field in an explicit analytical form as some quadratures.



The author expresses his gratitude to Professor M. A. Sumbatyan, Southern Federal University, Russia, for valuable comments. He would also like to notice that this work has been performed in frames of the project 9.5794.2017/8.9 under support of the Russian Ministry for Education and Science.


  1. 1.
    Ufimtsev, P.Y.: Fundamentals of the Physical Theory of Diffraction. Wiley, Hoboken, New Jercey (2007)Google Scholar
  2. 2.
    Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. SIAM, New York (1983)Google Scholar
  3. 3.
    Babich, V.M., Buldyrev, V.S.: Short-Wavelength Diffraction Theory. Springer, Heidelberg, Berlin (1989)Google Scholar
  4. 4.
    Shenderov, E.L.: Sound penetration through a rigid screen of finite thickness with apertures. Soviet Phys. Acoust. 16(2) (1970)Google Scholar
  5. 5.
    Sumbatyan, M.A., Scalia, A.: Equations of Mathematical Diffraction Theory. CRC Press, Boca Raton, Florida (2005)Google Scholar
  6. 6.
    Remizov, M.Y., Sumbatyan, M.A.: A semi-analytical method of solving problems of the high-frequency diffraction of elastic waves by cracks. J. Appl. Math. Mech. 77, 452–456 (2013)Google Scholar
  7. 7.
    Mittra, R., Lee, S.W.: Analytical Techniques in the Theory of Guided Waves. Macmillan, New York (1971)Google Scholar
  8. 8.
    Sumbatyan, M.A., Remizov, M.Y.: Asimotitic analysis in the anti-plane high-frequency diffraction by interface cracks. Appl. Math. Lett. 34, 72–75 (2014)CrossRefGoogle Scholar
  9. 9.
    Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)Google Scholar
  10. 10.
    Iovane, G., Lifanov, I.K., Sumbatyan, M.A.: On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics. Acta Mech. 162, 99–110 (2003)CrossRefGoogle Scholar
  11. 11.
    Chen, Z., Zhou, Y.F.: A new method for solving hypersingular integral equations of the first kind. Appl. Math. Lett. 24, 636–641 (2011)CrossRefGoogle Scholar
  12. 12.
    Fradkin, LJu, Stacey, R.: The high-frequency description of scatter of a plane compressional wave by an elliptic crack. Ultrasonics 50, 529–538 (2010)CrossRefGoogle Scholar
  13. 13.
    Rogoff, Z.M., Kiselev, A.P.: Diffraction at jump of curvature on an impedance boundary. J Wave Motion 33, 183–208 (2001)CrossRefGoogle Scholar
  14. 14.
    Gridin, D.: High-frequency asymptotic description of head waves and boundary layers surrounding the critical rays in an elastic half-space. J. Acoust. Soc. Am. 104, 1188–1197 (1998)CrossRefGoogle Scholar
  15. 15.
    Pal, S.C., Ghosh, M.L.: High frequency scattering of anti-plane shear waves by an interface crack Indian. J. Pure. Appl. Math. 21, 1107–1124 (1990)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Mathematics, Mechanics and Computer ScienceSouthern Federal UniversityRostov-on-DonRussia

Personalised recommendations