Skip to main content

Reflection and transmission of SH-waves at a corrugated interface between two laterally and vertically heterogeneous anisotropic elastic solid half-spaces

Abstract

This paper is concerned with the reflection and transmission coefficients of SH-waves at a corrugated interface between two anisotropic heterogeneous elastic solid half spaces. Both the half spaces are taken transversely isotropic and laterally and vertically heterogeneous. The Rayleigh’s method of approximation is adopted and expressions for reflection and transmission coefficients are obtained in closed form for the first-order approximation of the corrugation. In Rayleigh’s method, expressions in boundary conditions containing the function defining the corrugated boundary are expanded in Fourier series and unknown coefficients in the solutions are determined to any given order of approximation in terms of a small parameter characteristic of the boundary. The analytical expressions of these coefficients show that they depend upon corrugation of the interface and are strongly influenced by the anisotropy and heterogeneity of the half-spaces. Numerical computations are performed for the case of a particular corrugated interface: ζ =c cos k* x showing that the effect of heterogeneity on the reflection and transmission coefficients is minimum near the normal incidence and dominance of this effect increases with the angle of incidence. For incident wave striking at 45°, the effect of the corrugation is found significant on the reflection and transmission coefficients. The maximum effect of transverse isotropy on the reflection and transmission coefficients is observed at normal incidence when the values of the anisotropy parameters are 0.5 and 0.8 for the upper and lower half-spaces, respectively. The effect of frequency of the incident wave is observed on all reflected and refracted waves. The analytical expressions derived by Tomar and Saini (1997), Gupta (1987) and Asano (1960) are obtained as particular cases with our formulation.

References

  • Abubakar, I., Scattering of plane elastic waves at rough surfaces, Proc. Camb. Phil. Soc., 58, 136–157, 1962a.

    Article  Google Scholar 

  • Abubakar, I., Reflection and refraction of plane SH-waves at irregular interfaces, J. Phys. Earth, 10, 1–14, 1962b.

    Article  Google Scholar 

  • Abubakar, I., Reflection and refraction of plane SH-waves at irregular interfaces, J. Phys. Earth, 10, 15–20, 1962c.

    Article  Google Scholar 

  • Acharya, H. K., Reflection from the free surface of an inhomogeneous media, Bull. Seismol. Soc. Am., 60, 1101–1104, 1970.

    Google Scholar 

  • Adams, W. M., and Chung-Po-Chang, Wave propagation phenomenon at an irregular infinite interface, Part I, Bull. Seismol. Soc. Am., 54, 2209–2222, 1964.

    Google Scholar 

  • Aki, K. and K. L. Larner, Surface motion of layered medium having an irregular interface due to incident plane SH-waves, J. Geophys. Res., 75, 933–954, 1970.

    Article  Google Scholar 

  • Asano, S., Reflection and refraction of elastic waves at a corrugated boundary surface, Part I, The case of incidence of SH-wave, Bull. Earthq. Res. Inst., 38, 117–197, 1960.

    Google Scholar 

  • Asano, S., Reflection and refraction of elastic waves at a corrugated boundary surface, Part II, Bull. Earthq. Res. Inst., 39, 367–466, 1961.

    Google Scholar 

  • Asano, S., Reflection and refraction of elastic waves at a corrugated interface, Bull. Seismol. Soc. Am., 56, 201–221, 1966.

    Google Scholar 

  • Chattopadhyay, A. and A. Pal, Propagation of SH-waves in an inhomogeneous medium with irregular interface lying over an initially stressed elastic half space, Bul. Inst. Politeh. ‘Gheorghe Gheorghiu-Dej’, Bucuresti, Ser. Electroteh., 44(3), 47–56, 1982.

    Google Scholar 

  • Daley, P. F. and F. Hron, Reflection and transmission coefficients for transversely isotropic media, Bull. Seism. Soc. Am., 67(3), 661–675, 1977.

    Google Scholar 

  • Dunkin, J. W. and A. C. Eringen, The reflection of elastic waves from the wavy boundary of a half-space, Proceedings of the 4th U. S. National Congress on Applied Mechanics, University of California Press, Berkeley, 143–160, 1962.

    Google Scholar 

  • Gilbert, F. and L. Knopoff, Seismic scattering from topographic irregularities, J. Geophys. Res., 65, 3437–3444, 1960.

    Article  Google Scholar 

  • Gupta, R. N., Reflection of plane waves from a linear transition layer in liquid media, Geophysics, 30, 122–132, 1965.

    Article  Google Scholar 

  • Gupta, S. K., Reflection and refraction from curved interfaces: model-study, Geophys. Prospect., 26, 82–96, 1978.

    Article  Google Scholar 

  • Gupta, S., Reflection and transmission of SH-waves in laterally and vertically heterogeneous media at an irregular boundary, Geophysical Transactions, 33(2), 89–111, 1987. Henneke, E. G., Reflection-refraction of stress waves at a plane boundary between anisotropic media, J. Acoust. Soc. Am., 51, 210–217, 1972.

    Google Scholar 

  • Keith, C. M. and S. Crampin, Seismic body waves in anisotropic media: Reflection and refraction at a plane interface, Geophys. J. Roy. Astron. Soc., 49, 181–208, 1977.

    Article  Google Scholar 

  • Lay, T. and T. C. Wallace, Modern Global Seismology, Academic, San Diego, 1995.

    Google Scholar 

  • Malhotra, S., S. D. Chopra, and M. L. Gogna, Reflection and transmission of SH-waves from a laterally and vertically heterogeneous transition layer of finite thickness, Acta Geophysica Polonica, 30, 341–357, 1982.

    Google Scholar 

  • Mandal, B., Reflection and transmission properties of elastic wave on a plane interface for general anisotropic media, J. Acoust. Soc. Am., 90, 1106–1118, 1991.

    Article  Google Scholar 

  • Musgrave, M. J. P., Reflection and refraction of plane elastic waves at a plane boundary between aelotropic media, Geophys. J. Roy. Astron. Soc., 2, 406–418, 1960.

    Article  Google Scholar 

  • Rokhlin, S. I., T. K. Bolland, and L. Adler, Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media, J. Acoust. Soc. Am., 79, 906–918, 1986.

    Article  Google Scholar 

  • Ruger, A., P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry, Geophysics, 62, 713–722, 1997.

    Article  Google Scholar 

  • Saini, S. L. and S. J. Singh, Effect of anisotropy on the reflection of SH-waves at an interface, Geophys. Res. Bull., 15(2), 67–73, 1977a.

    Google Scholar 

  • Saini, S. L. and S. J. Singh, Reflection of SH-waves from an inhomogeneous half-space, Geophys. Res. Bull., 15(3), 1–12, 1977b.

    Google Scholar 

  • Sato, R., The reflection of elastic waves on corrugated surfaces, J. Seismol. Soc. Jpn., 8, 8–22, 1955.

    Google Scholar 

  • Shearer, P. M., Introduction to Seismology, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  • Sheriff, R. E. and L. P. Geldart, Exploration Seismology, 2nd ed., Cambridge University Press, Cambridge, 1995.

    Book  Google Scholar 

  • Singh, B. M., S. J. Singh, S. D. Chopra, and M. L. Gogna, On Love waves in laterally and vertically heterogeneous layered media, Geophys. J. Roy. Astron. Soc., 45, 357–370, 1976.

    Article  Google Scholar 

  • Singh, B. M., S. J. Singh, and S. D. Chopra, Reflection and refraction of SH-waves at the plane boundary between two laterally and vertically heterogeneous solids, Acta Geophys. Polonica, 36, 209–216, 1978.

    Google Scholar 

  • Slavin, L. M. and B. Wolf, Scattering of Love waves in a surface layer with an irregular boundary for the case of a rigid underlying half space, Bull. Seismol. Soc. Am., 60(3), 859–877, 1970.

    Google Scholar 

  • Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, NY, 1956.

    Google Scholar 

  • Sumner, J. H. and H. Deresiewicz, Waves in an elastic plate with an irregular boundary, PAGEOPH, 96, 106–125, 1972.

    Article  Google Scholar 

  • Tomar, S. K. and S. L. Saini, Reflection and refraction of SH-waves at a corrugated interface between two dimensional transversely isotropic half spaces, J. Phys. Earth, 45, 347–362, 1997.

    Article  Google Scholar 

  • Tomar, S. K., R. Kumar, and A. Chopra, Reflection/refraction of SH-waves at corrugated interface between transversely isotropic and visco-elastic solid half-spaces, Acta Geophysica Polonica, 50, 231–249, 2002.

    Google Scholar 

  • Yamada, T. and R. Sato, SH-wave propagation in medium having step-shaped discontinuity, J. Phys. Earth, 24, 105–130, 1976.

    Article  Google Scholar 

  • Zhang, R. C., L. Y. Zhang, and M. Shinozuka, Seismic waves in a laterally inhomogeneous layered medium: Theory, J. Appl. Mech., 64, 50–58, 1997.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. K. Tomar.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tomar, S.K., Kaur, J. Reflection and transmission of SH-waves at a corrugated interface between two laterally and vertically heterogeneous anisotropic elastic solid half-spaces. Earth Planet Sp 55, 531–547 (2003). https://doi.org/10.1186/BF03351786

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1186/BF03351786

Key words

  • Reflection
  • transmission
  • SH-wave
  • Rayleigh’s method
  • heterogeneity
  • anisotropy
  • corrugation