The asymptotic distribution of torsional eigenfrequencies of a spherical shell. I.

Abstract

In the two previous papers of this series (SATO and LAPWOOD, 1977a, b) we examined approximate methods for calculating eigenfrequencies of radial overtones of torsional oscillations of spherically symmetrical shells. For shells composed of uniform layers we were able to obtain an exact frequency equation, in terms of spherical Bessel functions, for which roots could be computed with any desired precision. They thus supplied a standard for the measurement of the accuracy of approximate methods.In applications to shells of two and three uniform layers, which were simple representations of an Earth with inner surfaces of discontinuity, we noted the presence of the solotone effect, which is the existence of recurring patterns of eigenfrequencies owing to internal reflection.In this paper we take up the analysis of the solotone effect, showing how it may be predicted from knowledge of the Shell structure, and how it may be interpreted in terms of ray theory. Applications to the same Earth-models as used before show that for them the theory of the solotone gives an excellent fit to the precisely computed eigenfrequencies. The pattern of eigenfrequencies proves to be very sensitive to changes in layer thickness, and thus offers the possibility of future use in determining the positions of surfaces of discontinuity within the mantle of the Earth.