Abstract
We consider the second-order differential equations ofP-SV motion in an isotropic elastic medium with spherical coordinates. We assume that in the medium Lamé's parameters λ, μ ∞r p and compressional and shear-wave velocities α, β ∞r, wherer is radial distance. With this regular heterogeneity both the radial functions appearing in displacement components satisfy a fourth-order differential equation which provides solutions in terms of exponential functions. We then consider a layered spherical earth in which each layer has heterogeneity as specified above. The dispersion equation of the Rayleigh wave is obtained using the Thomson-Haskel method. Due to exponential function solutions in each layer, the dispersion equation has similar simplicity, as in a flat-layered earth. The dispersion equation is further simplified, whenp=−2. We obtain numerical results which agree with results obtained by other methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alsop, L. E. (1963),Free Spheroidal Vibrations of the Earth at Very Long Periods. Part I. Calculations of Periods for Several Earth Models, Bull. Seismol. Soc. Am.53, 483–502.
Anderson, D. L. (1961),Elastic Wave Propagation in Layered Anisotropic Media, J. Geophys. Res.66, 2953–2963.
Bhattacharya, S. N. (1976),The Extension of Thomson-Haskell Method to Nonhomogeneous Spherical Layers, Geophys. J. R. Astr. Soc.47, 411–444.
Bhattacharya, S. N. (1978),Rayleigh Waves from a Point Source in a Spherical Medium with Homogeneous Layers, Bull. Seismol. Soc. Am.68, 231–238.
Biswas, N. N., andKnopoff, L. (1970),Exact Earth-flattening Calculation for Love Waves, Bull. Seismol. Soc. Am.60, 1123–1137.
Bolt, B. A., andDorman, J. (1961),Phase and Group Velocities of Rayleigh Waves in a Spherical Gravitating Earth, J. Geophys. Res.66, 2965–2981.
Gaulon, R., Jobert, N., Poupinet, G., andRoult, G. (1970),Application de la méthode de Haskell au calcul de la dispersion des ondes de Rayleigh sur un modele sphérique, Ann. Geophys.26, 1–8.
Haskell, N. A. (1953),The Dispersion of Surface Waves on Multilayered Media, Bull. Seismol. Soc. Am.43, 17–34.
Haskell, N. A. (1964),Radiation Pattern of Surface Waves from Point Sources in a Multilayered Medium, Bull. Seismol. Soc. Am.54, 377–393.
Hook, J. F. (1961), J. Acoust. Soc. Am.33, 302–313.
Hook, J. F. (1962),Contributions to a Theory of Separability of the Vector Wave Equation of Elasticity for Inhomogeneous Media, J. Acoust. Soc. Am.34, 946–953.
Landisman, M., Usami, T., Sato, Y., andMasse, R. (1970),Contribution of Theoretical Seismogram to the Study of Modes, Rays and the Earth, Rev. Geophys. Space Phys.8, 533–589.
Richards, P. G. (1974),Weakly Coupled Potentials of High-frequency Elastic Waves in Continuously Stratified Media, Bull. Seismol. Soc. Am.64, 1575–1588.
Saito, M. (1967),Excitation of Free Oscillations and Surface Waves by a Point Source in a Vertically heterogeneous Earth, J. Geophys. Res.,72, 3689–3699.
Schwab, F., andKnopoff, K. (1970),Surface Wave Dispersion Computations, Bull. Seismol. Soc. Am.61, 321–344.
Singh, S. J., andBen Menahem, A. (1969),Decoupling of the Vector Wave Equation of Elasticity for Radially Heteorogeneous Earth, J. Acoust. Soc. Am.46, 655–660.
Takeuchi, H., Saito, M., andKobayashi, N. (1962),Study of Shear Wave Velocity Distribution in the Upper Mantle by Mantle Rayleigh and Love Waves, J. Geophys. Res.67, 2831–2839.
Takeuchi, H., Dorman, J., andSaito, M. (1964),Partial Derivatives of Surface Wave Phase Velocity with Respect to Physical Parameter Changes within the Earth, J. Geophys. Res.69, 3429–3441.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arora, S., Bhattacharya, S.N. & Gogna, M.L. Rayleigh wave dispersion equation for a layered spherical earth with exponential function solutions in each shell. PAGEOPH 147, 515–536 (1996). https://doi.org/10.1007/BF00878842
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00878842