The vertical and horizontal displacements of a solid made up of one layer over a half-space, due to the presence of a vertical point-force acting beneath or at the free surface are calculated using a synthetic generalized ray expansion. Each term in the expansion represents a group of rays reflected the same number of times as P and S. The time dependence of the applied force at the source is a triangular function. Seismograms corresponding to the source term and for a half-space are also shown.
The behavior of the displacements is determined by the S-wave part of the motion. Some rays are clearly distinguishable, like the direct P or some of those following a totally reflected ray of the same type, like SP or SSP (denoted S2P). Also very prominent are the rays of type Sn arriving after diffracted ones at the free surface. It turns out, however, that the appearance of many portions of the seismograms is the result of the superposition of several generalized rays, contributing not only at their arrival but also later, at subsequent times.
The amplitude ratio of the P and S arrivals is shown to be related to the time dependence of the source: S appears to have a larger amplitude than P for short-duration sources.