Acta Mechanica

, Volume 230, Issue 1, pp 67–85 | Cite as

Effect of initial stress on the propagation and attenuation characteristics of Rayleigh waves

  • Santimoy Kundu
  • Manisha MaityEmail author
  • Deepak Kr. Pandit
  • Shishir Gupta
Original Paper


The present investigation deals with the mathematical modelling and analytical thinking to uncover the various facets of the propagation of Rayleigh waves in an Earth’s crustal layer. This work has been carried out when the wave is passing through a pre-stressed anisotropic layer of finite thickness, lying over a semi-infinite medium with void pores. The upper boundary plane of the crustal layer has been thought to be a free surface. Displacement components of the wave for both the media have been derived analytically. Appropriate boundary conditions have been well satisfied with the aid of displacement and stress factors in order to get the desired dispersion relation. A comparative study has been performed graphically taking anisotropic, orthotropic and isotropic strata, in order to show the impact of initial stress and thickness on the propagation characteristics of Rayleigh waves. The present work may establish a program to connect theoretical results with subject area applications.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Rayleigh, J.W.S.: On waves propagated along the plane surface of an elastic solid. Proc. Lond. Math. Soc. 17, 4–11 (1885)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ewing, W.M., Jardetsky, W.S., Press, F.: Elastic Waves in Layered Media. McGraw-Hill, New York (1957)CrossRefGoogle Scholar
  3. 3.
    Brekhovskikh, L.M.: Waves in Layered Media. Academic Press, New York (1960)Google Scholar
  4. 4.
    Kennet, B.L.: Seismic Wave Propagation in Stratified Media. Cambridge University Press, New York (1983)Google Scholar
  5. 5.
    Nayfeh, A.H.: Wave Propagation in Layered Anisotropic Media. Elsevier, Amsterdam (1995)zbMATHGoogle Scholar
  6. 6.
    Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland/American Elsevier, Amsterdam/New York (1973)zbMATHGoogle Scholar
  7. 7.
    Pilant, W.L.: Elastic Waves in the Earth. Elsevier, Amsterdam (1979)Google Scholar
  8. 8.
    Bullen, K.E., Bolt, B.A.: An Introduction to the Theory of Seismology. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  9. 9.
    Synge, J.L.: Elastic waves in anisotropic media. J. Math. Phys. 41, 323–334 (1957)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dutta, S.: Rayleigh wave propagation in a two layer anisotropic media. Pure Appl. Geophys. 60(1), 51–60 (1965)CrossRefGoogle Scholar
  11. 11.
    Sharma, M.D., Gogna, M.L.: Wave propagation in anisotropic liquid-saturated porous solids. J. Acoust. Soc. Am. 90, 1068–1073 (1991)CrossRefGoogle Scholar
  12. 12.
    Nayfeh, A.H.: The general problem of elastic wave propagation in multilayered anisotropic media. J. Acoust. Soc. Am. 89(4), 1521–1531 (1991)CrossRefGoogle Scholar
  13. 13.
    Vinh, P.C., Hue, T.T.T.: Rayleigh waves with impedance boundary conditions in anisotropic solids. Wave Motion 51, 1082–1092 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tanuma, K., Man, C.S., Chen, Y.: Dispersion of Rayleigh waves in weakly anisotropic media with vertically-inhomogeneous initial stress. Int. J. Eng. Sci. 92, 63–82 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pal, P.C., Kumar, S., Bose, S.: Propagation of Rayleigh waves in anisotropic layer overlying a semi-infinite sandy medium. Ain Shams Eng. J. 6, 621–627 (2015)CrossRefGoogle Scholar
  16. 16.
    Biot, M.A.: The influence of initial stress on elastic waves. J. Appl. Phys. 2, 522–530 (1940)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chattopadhyay, A., Mahata, N.P., Keshri, A.: Rayleigh wave in a medium under initial stresses. Acta Geophys. 34(1), 57–62 (1986)Google Scholar
  18. 18.
    Singh, B.: Wave propagation in a prestressed piezoelectric half-space. Acta Mech. 211(3), 337–344 (2010)CrossRefGoogle Scholar
  19. 19.
    Abd-Alla, A.M., Abo-Dahab, S.M., Hammad, H.A.H.: Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field. Appl. Math. Model. 35(6), 2981–3000 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sharma, A., Gupta, I.S.: Rayleigh waves in prestressed medium. Int. J. Research. Sci. Tech. 1(V), (2012)Google Scholar
  21. 21.
    Zhang, R., Pang, Y., Feng, W.: Propagation of Rayleigh waves in a magneto-electro-elastic half-space with initial stress. Mech. Adv. Mat. Struct. 21(7), 538–543 (2014)CrossRefGoogle Scholar
  22. 22.
    Pandit, D.K., Kundu, S., Gupta, S.: Propagation of Love waves in a pre-stressed Voigt-type viscoelastic orthotropic functionally graded layer over a porous half-space. Acta Mech. 228(3), 871–880 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nunziato, J.W., Cowin, S.C.: A non-linear theory of elastic material with voids. Arch. Ration. Mech. Anal. 72, 175–201 (1979)CrossRefGoogle Scholar
  24. 24.
    Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13(2), 125–147 (1983)CrossRefGoogle Scholar
  25. 25.
    Chandrasekharaiah, D.S.: Rayleigh–Lamb waves in an elastic plate with voids. J. Appl. Mech. 54, 509–512 (1987)CrossRefGoogle Scholar
  26. 26.
    Tomar, S.K.: Wave propagation in a micropolar elastic plate with voids. J. Vib. Cont. 11, 849–863 (2005)zbMATHGoogle Scholar
  27. 27.
    Iesan, D.: On a theory of thermo-viscoelastic materials with voids. J. Elast. 104(1), 369–384 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Singh, J., Tomar, S.K.: Plane waves in a rotating generalized thermo-elastic solid with voids. Int. J. Eng. Sci. Technol. 3(2), 34–41 (2011)MathSciNetGoogle Scholar
  29. 29.
    Vishwakarma, S.K., Gupta, S.: Rayleigh wave propagation: a case wise study in a layer over a half space under the effect of rigid boundary. Arch. Civil Mech. Eng. 14, 181–189 (2014)CrossRefGoogle Scholar
  30. 30.
    Biot, M.A.: Mechanics of Incremental Deformation. Wiley, New York (1965)CrossRefGoogle Scholar
  31. 31.
    Weiskopf, W.H.: Stresses in soils under a foundation. J. Franklin Inst. 239, 445–465 (1945)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ke, L.L., Wang, Y.S., Zhang, Z.M.: Love waves in an inhomogeneous fluid saturated porous layered half-space with linearly varying properties. Soil Dyn. Earthquake Eng. 26(6–7), 574–581 (2006)CrossRefGoogle Scholar
  33. 33.
    Rasolofosaon, P.N., Zinszner, B.E.: Comparison between permeability anisotropy and elasticity anisotropy of reservoir rocks. Geophysics 67(1), 230–240 (2002)CrossRefGoogle Scholar
  34. 34.
    Puri, P., Cowin, S.C.: Plane waves in linear elastic materials with voids. J. Elast. 15(2), 167–183 (1985)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Santimoy Kundu
    • 1
  • Manisha Maity
    • 1
    Email author
  • Deepak Kr. Pandit
    • 2
  • Shishir Gupta
    • 1
  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.Department of Basic Science and HumanitiesUniversity of Engineering and ManagementKolkataIndia

Personalised recommendations