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 r1 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 r<r1, and by a logarithmic infinity for r>r1. The coefficient of the logarithmic term is zero both at r=r1 and at large ranges, having a sharp maximum at r=1.004r1. There is no Rayleigh wave at r<r1. 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.

The Use of Singular Integrals in Wave Propagation Problith Application to the Point Source in a Semi-Infinite Elastic Medium

“…Thus with w = 0 . These simple expressions are the initial displacements on the surface linked with the arrival of P and S waves; they agree with the results of Pekeris (1957) for the horizontal components of displacement, but for the vertical component they do not. (His initial P displacement lacks the factor 2 22½ (I-h /cI t )…”

“…These limits have been mentioned elsewhere (Ewing et al, 1957), and they differ from the empirical result given by Pekeris (1957).…”

“…The method to be described here will, in its generality, involve us in the calculation of known results for the fixed source. In the process we find that the results of Pekeris and Lifson (1957), for the vertical force are incorrect in the vertical displacement component. The method is of special interest because specific parts of the field are picked out without ambiguity; not only can we pick out the singular parts of the field for the general point source, but we can pick out the stages in the development of the head wave field as well.…”

Transient response of a thermoelastic half‐space to mechanical and thermal buried sources

“…at the free surface [45]. When the S-wave reaches at the free surface, it is reflected partially as S wave and partially as P wave, the latter at r = H/ √ 2, propagates in the horizontal direction and owes its existence entirely to diffraction (see Fig.…”

“…4). For the points further away than H/ √ 2, in addition to the direct S-wave, there exists another wave known as the SP wave, which travels on a part of its trajectory with the shear wave velocity and on the other part, which is along the horizontal surface with dilatational wave velocity [45]. For the points near the source (r < H/ √ 2) the P -and S-waves cause finite jumps at the amplitude of vertical displacement.…”