Introduction

After the homogeneous infinite space, the next two simplest configurations are a homogeneous half-space with a free surface and two homogeneous half-spaces (or media, for short) with different elastic properties. The first case can be considered as a special case of the second one with one of the media a vacuum. In either case the boundary between the two media constitutes a surface of discontinuity in elastic properties that has a critical effect on wave propagation. To simplify the problem we will assume plane boundaries and wave fronts. Although in the Earth neither the wave fronts nor the boundaries satisfy these assumptions, they are acceptable approximations as long as the seismic source is sufficiently far from the receiver and/or the wavelength is much shorter than the curvature of the boundary. In addition, the case of spherical wave fronts can be solved in terms of plane wave results (e.g., Aki and Richards, 1980). Therefore, the theory and results described here have a much wider application than could be expected by considering the simplifying assumptions. For example, they are used in teleseismic studies, in the generation of synthetic seismograms using ray theory, and in exploration seismology, particularly in amplitude-versus-offset (AVO) studies.

The interaction of elastic waves with a boundary has a number of similarities with the interaction of acoustic and electromagnetic waves, so that it can be expected that a wave incident on a boundary will generate reflected and transmitted waves (the latter only if the other medium is not a vacuum).