THE ASYMPTOTIC DISTRIBUTION OF TORSIONAL EIGENFREQUENCIES OF A SPHERICAL SHELL. II

Abstract

The detailed pattern of eigenfrequencies of radial overtones of torsional modes of oscillation of a radially-symmetric layered sphere appears to be intimately related to the set of discontinuities of the parameters μ(γ) (rigidity) and ρ(γ) (density).
In order to elucidate this relationship, we explored in Part I of this series (SATO and LAPWOOD, 1977) the asymptotic approximations available for the discussion of the frequency equation for radial overtones corresponding to a given Legendre parameter for two simple models-a uniform shell and a shell composed of two uniform layers.
In this paper we extend the theory to a general shell of l uniform layers, and obtain asymptotic approximations to the frequency equation by means of (a) Stokes-type approximations for spherical Bessel functions, (b) Green-type approximations, and (c) Sturm-Liouville theory. We show how (b) and (c) take both Earth-curvature and internal reflexions into account.
The theory is applied to an Earth-model with a three-layer shell, which is obtained by averaging from PEM-A of DZIEWONSKI et al. (1975). This model has discontinuities a approximate depths 400km and 600km. The relative accuracy of three approximations is explored, and the existence of a solotone effect is exhibited.