Abstract
The present paper contributes in studying the phase velocities of P- and S-waves in a half space subjected to a compressive initial stress and gravity field. The density and acceleration due to gravity vary quadratically along the depth. The dispersion equation is derived in a closed form. It is shown that the phase velocities depend not only on the initial stress, gravity, and direction of propagation but also on the inhomogeneity parameter associated with the density and acceleration due to gravity. Various particular cases are obtained, and the results match with the classical results. Numerical investigations on the phase velocities of P- and S-waves against the wave number are made for various sets of values of the material parameters, and the results are illustrated graphically. The graphical user interface model is developed to generalize the effect.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ewing, W. M., Jardetzky, W. S., and Press, F. Elastic Waves in Layered Media, McGraw-Hill, New York (1957)
Bullen, K. E. An Introduction to the Theory of Seismology, Cambridge University Press, London (1963)
Bath, M. Mathematical Aspects of Seismology, Elsevier Publishing, Amsterdam (1968)
Miklowitz, J. Elastic wave propagation. Applied Mechanics Survey, Spartan Books, Michigan (1966)
Achenbach, J. D. Wave Propagation in Elastic Solids, North-Holland, Amsterdam (1973)
Kalski, H. Stress Waves in Solids, Dover Publications, New York (1963)
Aki, K. and Larner, K. L. Surface motion of a layered medium having an irregular interface due to incident plane SH-waves. J. Geophys. Res., 40, 933–954 (1970)
Bard, P. Y. and Bouchon, M. The seismic response of sediment-filled valleys, part 1: the case of incident SH waves. Bulletein of the Seismological Society America, 70, 1263–1286 (1980)
Babich, S. I. and Guz, A. N. Plane dynamic problems for elastic incompressible bodies with initial stresses. Journal of Applied Mathematics and Mechanics, 46, 197–204 (1982)
Alekseev, A. S. and Mikhailenko, B. G. Solution of dynamic problems of elastic wave propagation in inhomogeneous media by a combination of partial separation of variables and finite difference methods. Geophysical Journal International, 48, 161–172 (1980)
Borcherdt, R. D. Reflection and refraction of type II S-waves in elastic and anelastic media. Bulletin of the Seismological Society of America, 67(1), 43–67 (1977)
Borcherdt, R. D. Reflection and refraction of general P- and type I S-waves in elastic and an elastic solid. Geophysical Journal International, 70, 621–638 (1982)
Chattopadhyay, A., Bose, S., and Chakraborty, M. Reflection of elastic wave under initial stress at a free surface, P- and SV -motion. Journal of Acoustical Society of America, 72, 255–263 (1982)
Chattopadhyay, A., Kumari, P., and Sharma, V. K. Reflection and refraction of three dimensional plane quasi-P waves at a corrugated surface between distinct triclinic elastic half spaces. International Journal on Geomathematics, 2(2), 219–253 (2011)
Dey, S. and Dutta, A. P and S waves in a medium under initial stresses and under gravity. Indian Journal of Pure Applied Mathematics, 15(7), 795–808 (1984)
Biot, M. A. Mechanics of Incremental Deformation, John Wiley and Sons, New York (1965)
Dey, S. and Mahto, P. Effective elastic constants in an medium under initial stress and prediction of liquid core inside a gravitating medium. Bulletin of Calcutta Mathematical Society, 58, 107–114 (1993)
Gubbins, D. Seismology and Plate Tectonics, Cambridge University Press, Cambridge/New York (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the Research Fellow of Indian School of Mines in Dhanbad (No. 2010DR0016)
Rights and permissions
About this article
Cite this article
Gupta, S., Vishwakarma, S.K. Propagation of P- and S-waves in initially stressed gravitating half space. Appl. Math. Mech.-Engl. Ed. 34, 847–860 (2013). https://doi.org/10.1007/s10483-013-1712-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-013-1712-7