A Comparative Analysis of Wave Properties of Finite and Infinite Cascading Arrays of Cracks

  • Vitaly V. Popuzin
  • M. Y. Remizov
  • Mezhlum A. SumbatyanEmail author
  • Michele Brigante
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 109)


Reflection and transmission coefficients in the problems of the normal plane wave incidence on the system of finite and infinite periodic arrays of cracks in an elastic body are determined. We propose a method permitting to solve the scalar diffraction problem for both single crack and any finite number of cracks with arbitrary lattice geometry. Under the condition of one-mode frequency regime the problem is reduced to a discretization of the basic integral equation holding on the boundary of the scatterers located in one horizontal waveguide. A semi-analytical method developed earlier for diffraction problems on infinite periodic crack arrays permits a comparative analysis of the properties of the main external parameters for a finite periodic system of cracks, where the solution of the boundary integral equations is numerically constructed, and we obtain explicit analytical representations for the wave field at the boundary of the obstacles. The analysis of the properties of the scattering coefficients depending on the physical parameters is carried out for three diffraction problems: a finite periodic system in a scalar formulation, an infinite periodic system in a scalar formulation, an infinite periodic system in a plane problem of the elasticity theory.



The present work is performed within the framework of the Project no. 15-19-10008-P of the Russian Science Foundation (RSCF).


  1. 1.
    Guenneau, S., Craster, R.V.: Acoustic Metamaterials Negative Refraction, Imaging, Lensing and Cloaking. Springer, Dordrecht, Heidelberg, New York, London (2013)Google Scholar
  2. 2.
    Deymier, P.A.: Acoustic Metamaterials and Phononic Crystals. Springer-Verlag, Berlin, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Twersky, V.: Multiple scattering of sound by a periodic line of obstacles. J. Acoust. Soc. Am. 53(1), 329–338 (1973)CrossRefGoogle Scholar
  4. 4.
    Shenderov, E.L.: Transmission of sound through a perforated screen of finite thickness. Acoust. Phys. 16(2), 295–304 (1970)Google Scholar
  5. 5.
    Achenbach, J.D., Li, Z.L.: Reflection and transmission of scalar waves by a periodic array of screens. Wave Motion 1986(8), 225–234 (1986)CrossRefGoogle Scholar
  6. 6.
    Miles, J.W.: On Rayleigh scattering by a grating. Wave Motion 4, 285–292 (1982)CrossRefGoogle Scholar
  7. 7.
    Scarpetta, E.: In-plane problem for wave propagation through elastic solids with a periodic array of cracks. Acta Mech. 154, 179–187 (2002)CrossRefGoogle Scholar
  8. 8.
    Sumbatyan, M.A., Remizov, MYu.: On the theory of acoustic metamaterials with a triple-periodic system of interior obstacles. Adv. Struct. Mater. 41, 19–33 (2017)CrossRefGoogle Scholar
  9. 9.
    Zarrillo, G., Aguiar, K.: Closed-form low frequency solutions for electro-magnetic waves through a frequency selective surface. IEEE Trans. Antennas Prop. AP-35(12), 1406–1417 (1988)CrossRefGoogle Scholar
  10. 10.
    Sumbatyan, M.A., Chupahin, A.A.: Plane wave propagation through an elastic medium with a periodic system of volumetric defects. Russian Izvestiya. North-Caucas. Region. Nat. Sci. 4, 37–38 (1999). (in Russian)Google Scholar
  11. 11.
    Scarpetta, E., Sumbatyan, M.A.: Wave propagation through elastic solids with a periodic array of arbitrarily shaped defects. Math. Comp. Model. 37(1, 2), 19–28 (2003)CrossRefGoogle Scholar
  12. 12.
    Angel, Y.C., Bolshakov, A.: In-plane waves in an elastic solid containing a cracked slab region. Wave Motion 31, 297–315 (2000)CrossRefGoogle Scholar
  13. 13.
    Angel, Y.C., Achenbach, J.D.: Harmonic waves in an elastic solid containing a doubly periodic array of cracks. Wave Motion 9, 377–385 (1987)CrossRefGoogle Scholar
  14. 14.
    Mykhaskiv, V.V., Zhbadynskyi, I.Y., Zhang, C.: Dynamic stresses due to time-harmonic elastic wave incidence on doubly periodic array of penny-shaped cracks. J. Math. Sci. 203, 114–122 (2014)CrossRefGoogle Scholar
  15. 15.
    Scarpetta, E., Sumbatyan, M.A.: On wave propagation in elastic solids with a doubly periodic array of cracks. Wave Motion 25, 61–72 (1997)CrossRefGoogle Scholar
  16. 16.
    Remizov, M.Y.: Low-frequency penetration of elastic waves through a periodic array of cracks. Vestn. Don State Tech. Univ. 1(88), 18–27 (2017). (in Russian)CrossRefGoogle Scholar
  17. 17.
    Remizov, M.Y., Sumbatyan, M.A.: 3-D one—mode penetration of elastic waves through a doubly periodic array of cracks. Math. Mech. Solids 23(4), 636–650 (2018)CrossRefGoogle Scholar
  18. 18.
    Glazanov, V.E.: Diffraction of a plane longitudinal wave by a lattice of cylindrical cavities in an elastic medium. Acoust. Phys. 13(3) (1967)Google Scholar
  19. 19.
    Kljukin, I.I., Chabanov, V.E.: Sound diffraction on a plane grating of cylinders. Acoust. Phys. 20(6) (1974)Google Scholar
  20. 20.
    Popuzin, V.V., Zotov, V.M., Sumbatyan, M.A.: Theoretical and experimental study of an acoustically active material containing a doubly-periodic system of cylindrical holes. Springer Proceedings in Physics, Heidelberg, New York, Dordrecht, London (2017)Google Scholar
  21. 21.
    Datta, S.K.: Diffraction of plane elastic waves by ellipsoidal inclusions. J. Acoust. Soc. Am. 61, 1432–1437 (1977)CrossRefGoogle Scholar
  22. 22.
    Willis, J.R.: A polarization approach to the scattering of elastic waves—II. J. Mech. Phys. Solids 28, 307–327 (1980)CrossRefGoogle Scholar
  23. 23.
    Kuznetsov, S.V.: A direct version of the method of boundary integral equations in the theory of elasticity. J. Appl. Math. Mech. 56(5), 617–622 (1992)CrossRefGoogle Scholar
  24. 24.
    Yang, Ch., Achenbach, J.D.: Time domain scattering of elastic waves by a cavity, represented by radiation from equivalent body forces. Int. J. Eng. Sci. 115, 43–50 (2017)CrossRefGoogle Scholar
  25. 25.
    Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic material. Science 289(5485), 1734–1736 (2000)CrossRefGoogle Scholar
  26. 26.
    Craster, R.V., Guenneau, S.: Acoustic Metamaterials. Springer Series in Materials Science, vol. 166. Springer, Dordrecht (2013)Google Scholar
  27. 27.
    Huang, H.H., Sun, C.T., Huang, G.L.: On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci. 47, 610–617 (2009)CrossRefGoogle Scholar
  28. 28.
    Kriegsmann, G.A.: Scattering matrix analysis of a photonic Fabry-Perot resonator. Wave Motion 37, 43–61 (2003)CrossRefGoogle Scholar
  29. 29.
    Scarpetta, E., Sumbatyan, M.A.: On the oblique wave penetration in elastic solids with a doubly periodic array of cracks. Q. Appl. Math. 58, 239–250 (2000)CrossRefGoogle Scholar
  30. 30.
    Sneddon, I.N., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. Wiley, London (1969)Google Scholar
  31. 31.
    Belotserkovsky, S.M., Lifanov, I.K.: Method of Discrete Vortices. CRC Press, Boca Raton (1992)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vitaly V. Popuzin
    • 1
  • M. Y. Remizov
    • 1
  • Mezhlum A. Sumbatyan
    • 1
    Email author
  • Michele Brigante
    • 2
  1. 1.Southern Federal UniversityRostov-on-DonRussian Federation
  2. 2.University of Naples—Federico IINaplesItaly

Personalised recommendations