This chapter begins with coverage of the quantitative concepts used to describe the deformation of solids by seismic waves, namely the concepts of stress, strain, and dilatation. This is followed by the derivation of equations for describing seismic wave motion in the subsurface, namely, the equation of motion, conservation of energy, kinetic and strain-energy density, intensity or energy flux, the stress–strain relation, isotropy, hydrostatic stress, elastic constants (which are related to the nature of the medium in which waves travel), the wave equations, compressional and shear waves, plane harmonic waves, displacement potentials, Helmholtz equations, near-field and far-field waves, mean values, and the acoustic wave equation. The chapter ends with examples that discuss seismic waves produced by a buried explosive charge and by a directed point force, and discussions of the moment tensor and apparent velocities.

The subsurface of the Earth consists typically of media that are not homogeneous, but rather heterogeneous (or inhomogeneous), such as sequences of homogeneous layers where each layer has a different density and wave velocity, or zones where the wave velocity and medium parameters vary smoothly with position. This chapter looks at the theory of how seismic waves propagate in such heterogeneous media. In addition, coverage includes applications of the theory to well surveys and logs (tools used in exploration seismology), the Wiechert–Herglotz method for computing how the seismic wave velocity varies with position from measurements of seismic wave travel times, the eikonal equation (an equation for computing wave travel times), zero-offset ray tracing (a relatively simple technique for roughly estimating the nature of the seismic data that might be recorded in regions of complex subsurface structure), diffracted seismic waves, acoustic waves in heterogeneous materials, and ray equations (for computing the paths of seismic waves in heterogeneous media).