Basic Study on the Oscillation of a Sphere.

Abstract

Summary The propagation of an impulsive torsional disturbance excited by a localized stress on the surface of a sphere consisting of a homogeneous mantle surrounding a liquid core is expressed as the sum of normal mode solutions. The formulas presented can be used for the calculation of the disturbance on the surface of a sphere having any arbitrary radial distribution of material. Since the medium is a finite bounded body, the normal modes offer a complete description of the propagating disturbance. The azimuthal component of displacement at various points on the surface is calculated numerically as a function of time by summing the spectral contributions from 360 normal modes equally distributed among the first six radial modes. The surface waves, which are similar to those observed from earthquakes, are caused primarily by the fundamental radial mode, while the body waves are primarily related to the higher radial modes. The liquid core causes a shadow zone for the S wave; travel times for diffracted S correspond to the simple geometrical theory of travel to and from the curved core boundary along the tangent paths. The core-mantle boundary also causes the ScS phases which may be multiply reflected from the free surface.