Research Note Asymptotic Structure in Eigenfrequencies of Spheroidal Normal Modes of the Earth

Summary The modes of the spheroidal oscillations ,,S, of the Earth can, for small 1, be classified according to the body wave modes PKIKP (compressional oscillations in the whole Earth), (ScS), (shear oscillations in the mantle) and J (shear oscillations in the inner core). Analysis of the asymptotic behaviour of the eigenperiods according to this classification shows that the asymptotic structure for modes of PKIKP type is perturbed by the presence of the core-mantle and inner-outer core boundaries, whereas that for modes of (ScS), type depends strongly on upper mantle structure. The inner core was virtually homogeneous in the models analysed, and this was reflected in an absence of perturbations in the asymptote of the J mode. Estimates of the radial travel times of each wave type can be derived from the asymptotic eigenperiods. The radial modes ,,So are all of PKIKP type, and the asymptotic structure of observed eigenperiods of these modes is clearly perturbed by the presence of the core discontinuities, thus con6rming the existence of the solotone effect analysed theoretically by McNabb et al. Anderssen, Cleary & Osborne (1974) have shown recently that the asymptotic behaviour of the Earth's torsional oscillations T,, for small I and large n, can be used to estimate the radial travel time of (ScS), waves. Subsequently, Anderssen & Cleary (1974) demonstrated from model experiments that the presence of discontinuities in the Earth's mantle can produce perturbations in the asymptotic structure of torsional oscillations. Lapwood (1975) and McNabb, Anderson & Lapwood (1 975) have analysed these conclusions theoretically. In particular, the latter showed that the asymptotic structure of the eigenvalues of a related class of Sturm-Liouville problems with discontinuous coefficients can exhibit a solotone efect in the form of an irregular pattern of eigenvalues which is repeated indefinitely. In order to extend these results to the Earth's spheroidal oscillations ,,S,, we consider first the physical meaning of the leading terms in the asymptotic formula