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Published February 1978 | Published
Journal Article Open

Normal modes of a laterally heterogeneous body: A one-dimensional example


Various methods, including first- and second-order perturbation theory and variational methods have been proposed for calculating the normal modes of a laterally heterogeneous earth. In this paper, we test all three of these methods for a simple one-dimensional example for which the exact solution is available: an initially homogeneous "string" in which the density and stiffness are increased in one half and decreased in the other by equal amounts. It is found that first-order perturbation theory (as commonly applied in seismology) yields only the eigenvalues and eigenfunctions for a string with the average elastic properties; second-order perturbation theory is worse, because the eigenfunction is assumed to be the original eigenfunction plus small correction terms, but actually may be almost completely different. The variational method (Rayleigh-Ritz), using the unperturbed modes as trial functions, succeeds in giving correct eigenvalues and eigenfunctions even for modes of high-order number. For the example problem only the variational solution correctly yields the transient solution for excitation by a point force, including correct amplitudes for waves reflected by and transmitted through the discontinuity. Our result suggests but does not demonstrate, that the variational method may be the most appropriate method for finding the normal modes of a laterally heterogeneous earth model, particularly if the transient solution is desired.

Additional Information

Copyright © 1978, by the Seismological Society of America. Manuscript received September 2, 1977. We thank Jon Claerbout, Yoshio Fukao, Hiroo Kanamori, and Emile Okal for helpful discussions. Gene Golub, Herb Keller, and Dan Kosloff gave us useful advice on the matrix eigenvalue problem. We also thank Tony Dahlen, Raul Madariaga, and Robert Anderssen for constructive comments on an earlier draft. This research was supported by the National Science Foundation under Grants EAR76-14262 and EAR 77-14675. Seth Stein was supported by a Fellowship from the Fannie and John Hertz Foundation.

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