The Transmission of a Rayleigh Pulse round a corner

Abstract

Lamb, in his classical paper of 1904, found the form of the Rayleigh pulse that travels over the surface of an elastic half‐space, which has been struck along a line. Let such a pulse travel on one face of an elastic quarter‐space, the crest‐line being parallel to and travelling towards the edge. We enquire what form the Rayleigh pulse will have after passing round the corner. The problem is taken as two‐dimensional; the elastic solid fills the positive quadrant xOy and the incident Rayleigh pulse arises from a pressure p(x, t)= ‐ Qδ(x ‐a)φ(t) applied on the face y= o. The differential equations and boundary conditions are reduced to operational form by the use of a Fourier transform with respect to time and Laplace transforms with respect to x and y. Difficulties arise from our lack of knowledge of the displacements at the surfaces x= o and y= o, but solution of the problem is reduced to that of two simultaneous integral equations. An iterative solution can be found, and from this we isolate the parts which describe the incident Rayleigh pulse (identical with Lamb's solution) and the pulse transmitted round the corner. It is found that the form of the pulse is greatly changed, in ways depending on the values of the velocities of P, S and Rayleigh waves. Whereas the dispacements u and v on y= o have the shape of φ(t) and its allied function (Hilbert transform) φ(t) respectively, each of u and v on x= o is given by a linear combination of φ and φ. Since φ may differ greatly from φ in form, the change in shape of each component of displacement when it turns the corner may be very marked. Copyright © 1961, Wiley Blackwell. All rights reserved