The pattern of eigenfrequencies of radial overtones which is predicted for a specified Earth-model

In 1974 Anderssen and Cleary examined the distribution of eigenfrequencies of radial overtones in torsional oscillations of Earth-models. They pointed out that according to Sturm-Liouville theory this distribution should approach asymptotically, for large overtone number m, the value nnz/y, where y is the time taken by a shear-wave to travel along a radius from the core-mantle interface to the surface, provided elastic parameters vary continuously along the radius. They found that, for all the models which they considered, the distributions of eigenfrequencies deviated from the asymptote by amounts which depended on the existence and size of internal discontinuities. Lapwood (1975) showed that such deviations were to be expected from Sturm-Liouville theory, and McNabb, Anderssen and Lapwood (1976) extended Sturm-Liouville theory to apply to differential equations with discontinuous coefficients. Anderssen (1977) used their results to show how to predict the pattern of deviations —called by McNabb et al. the solotone effect— for a given discontinuity in an Earth-model. Recently Sato and Lapwood (1977), in a series of papers which will be referred to here simply as I, II, III, have explored the solotone effect for layered spherical shells, using approximations derived from an exacttheory which holds for uniform layering. They have shown how the form of the pattern of eigenfrequencies, which is the graph of S — YMUJI/N — m against m, where ,„CJI is the frequency of the m"' overtone in the I"' (Legendre) mode of torsional oscillation, is determined as to periodicity (or quasi-periodicity) by the thicknesses and velocities of the layers, and as to amplitude by the amounts of the discontinuities (III). The pattern of eigenfrequencies proves to be extremely sensitive to small changes in layer-thicknesses in a model. In this paper we examine a proposed Earth-model with six surfaces of discontinuity between core boundary and surface, and predict its pattern of eigenfrequencies. When seismological observations become precise enough, and can be subjected to numerical analysis refined enough, to identify the radial overtones for large m, this pattern of eigenfrequencies will prove to be a severe test for any proposed model, including he one which we discuss below.