Reflection and transmission of SH-waves at a corrugated interface between two laterally and vertically heterogeneous anisotropic elastic solid half-spaces
Earth, Planets and Space volume 55, pages 531–547 (2003)
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.
Abubakar, I., Scattering of plane elastic waves at rough surfaces, Proc. Camb. Phil. Soc., 58, 136–157, 1962a.
Abubakar, I., Reflection and refraction of plane SH-waves at irregular interfaces, J. Phys. Earth, 10, 1–14, 1962b.
Abubakar, I., Reflection and refraction of plane SH-waves at irregular interfaces, J. Phys. Earth, 10, 15–20, 1962c.
Acharya, H. K., Reflection from the free surface of an inhomogeneous media, Bull. Seismol. Soc. Am., 60, 1101–1104, 1970.
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.
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.
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.
Asano, S., Reflection and refraction of elastic waves at a corrugated boundary surface, Part II, Bull. Earthq. Res. Inst., 39, 367–466, 1961.
Asano, S., Reflection and refraction of elastic waves at a corrugated interface, Bull. Seismol. Soc. Am., 56, 201–221, 1966.
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.
Daley, P. F. and F. Hron, Reflection and transmission coefficients for transversely isotropic media, Bull. Seism. Soc. Am., 67(3), 661–675, 1977.
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.
Gilbert, F. and L. Knopoff, Seismic scattering from topographic irregularities, J. Geophys. Res., 65, 3437–3444, 1960.
Gupta, R. N., Reflection of plane waves from a linear transition layer in liquid media, Geophysics, 30, 122–132, 1965.
Gupta, S. K., Reflection and refraction from curved interfaces: model-study, Geophys. Prospect., 26, 82–96, 1978.
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.
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.
Lay, T. and T. C. Wallace, Modern Global Seismology, Academic, San Diego, 1995.
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.
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.
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.
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.
Ruger, A., P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry, Geophysics, 62, 713–722, 1997.
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.
Saini, S. L. and S. J. Singh, Reflection of SH-waves from an inhomogeneous half-space, Geophys. Res. Bull., 15(3), 1–12, 1977b.
Sato, R., The reflection of elastic waves on corrugated surfaces, J. Seismol. Soc. Jpn., 8, 8–22, 1955.
Shearer, P. M., Introduction to Seismology, Cambridge University Press, Cambridge, 1999.
Sheriff, R. E. and L. P. Geldart, Exploration Seismology, 2nd ed., Cambridge University Press, Cambridge, 1995.
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.
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.
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.
Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, NY, 1956.
Sumner, J. H. and H. Deresiewicz, Waves in an elastic plate with an irregular boundary, PAGEOPH, 96, 106–125, 1972.
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.
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.
Yamada, T. and R. Sato, SH-wave propagation in medium having step-shaped discontinuity, J. Phys. Earth, 24, 105–130, 1976.
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.
About this article
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