Motion of the Surface of a Uniform Elastic Half-Space Produced by a Buried Pulse

Abstract

An exact solution is obtained for the motion of the surface of a uniform elastic half‐space due to the application at a depth H below the surface of a concentrated vertical force. The time‐variation of the applied force is assumed to be represented by the Heaviside unit function. The solution for the horizontal and vertical components of displacement cast in the form of single integrals over a fixed range, and these have been evaluated on the electronic computer of the Weizmann Institute (WEIZAC). The assumed source emits both S waves and P waves. Beyond a distance r 1 from the epicenter, which is equal to H/√ in the case λ = μ, the original S wave is converted on reaching the surface into a diffracted SP wave traveling along the surface. At large ranges, the SP phase is more pronounced than the P phase. The S phase is marked by a finite jump for rr 1. The coefficient of the logarithmic term is zero both at r=r 1 and at large ranges, having a sharp maximum at r=1.004r 1. There is no Rayleigh wave at r<r 1. At large ranges (r/H≫1) the solution, as a function of the reduced time τ = ct/R, approaches the form of the solution for the surface pulse.