A finite difference equation formulation for the equations of elasticity is presented and applied to the problem of a layered half-space with a buried point source emitting a compressional pulse. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The results for a specific case are compared with those found by a completely different method in order to check the validity of the finite difference methods. The agreement is excellent. The effect of different mesh sizes on the theoretical seismograms is studied next and a suitable grid system selected for the applications that follow. The development of Rayleigh waves on the surface of a half-space and the change of the Rayleigh wave with depth and pulse width are examined. The problem of a layered half-space with a high velocity bottom is considered and the refraction arrivals on the surface and on the interface are studied. The problem of interface waves on the surface separating two semiinfinite media is also examined. Interface waves are found when the physical parameters lie both inside and outside the region determined by the Stoneley equation. Finally, a series of theoretical seismograms for a layered half-space showing the variation of the surface waves as a function of depth and of the density in the lower medium is presented.