The propagation of earthquake waves

Abstract

1. A complete understanding of the propagation of earthquake disturbances demands a solution of the problem of the vibrations of an elastic sphere in which the mutual gravitational attraction of the parts is taken into account, as well as the variation of the physical constants of the earth’s material with distance from the centre. The classical solution of Lamb applied only to a sphere in which gravitation is supposed negligible, and in which the physical constants have uniform values throughout. In a subsequent discussion of certain aspects of the general problem, I tried to take gravitation into account, but was only able to make progress by introducing the quite artificial assumption of a permanent field of force exactly annulling the gravitational field of the sphere in its undisturbed state. In a valuable critical note on this paper, the late Lord Rayleigh indicated a method which opened up possibilities of progress while avoiding this very artificial restriction. In the present note, I have availed myself of Lord Rayleigh’s suggestion, and have treated the problem in certain of its aspects which seem to have a direct seismological importance. A summary and discussion of results will be found at the end (§ 18). 2. We assume the earth to be spherical in shape and to be arranged in spherical layers, so that all the physical constants are functions of a single variable, r , the distance from the centre. In the absence of earthquake disturbances, the stress system at any point is assumed to reduce to a simple hydrostatic pressure, p . This will be connected with the density and the ρ gravitational potential V by the usual equations ∂ p /∂ x = ρ ∂V/∂ x , etc. in which p, ρ and V are functions of r only.