Solutions for problems involving wave propagation in a semi-infinite half space are of interest for interpreting measurements of radiation fields at locations near or on the free surface. Solutions to these problems as derived for elastic media have formed the basis for the initial interpretation of seismograms and resultant inferences concerning the internal structure of the Earth.

Analytic solutions and corresponding numerical examples for problems involving general SI, P, and SII waves incident on the free surface of a viscoelastic half space are presented in this chapter (Borcherdt, 1971, 1988; Borcherdt and Glassmoyer, 1989; Borcherdt et al., 1989). Closed-form expressions for displacement and volumetric strain are included to facilitate understanding and interpretation of measurements as might be detected on seismometers and volumetric strain meters at or near the free surface of a viscoelastic half space.

The procedures to solve the reflection–refraction problems for a general SI, P, or SII wave incident on a free surface are analogous to those for the corresponding problems for a welded boundary. For brevity, many of the expressions and results in medium V for a welded boundary applicable to the free-surface problems will be referred to here, but not rewritten.

Boundary-Condition Equations

Solutions of the equations of motion for problems of general P, SI, and SII waves incident and reflected from the surface of a viscoelastic half space are specified by (4.2.1) through (4.2.45) with respect to the coordinate system illustrated in Figure (4.1.3), where medium V′ is assumed to be a vacuum.