Using reciprocity to derive the far field displacements due to buried sources and scatterers

It is shown that elastodynamic reciprocity provides a simpler approach for deriving the far-field displacements due to buried (sub-surface) sources in a half-space, compared with integral transform techniques. The auxiliary fields employed in this approach are the fields associated with the reflection of plane waves of the three possible polarisations, and the required far field can be expressed in terms of these well-known auxiliary fields. The crucial step in this approach is to evaluate a surface integral involving cross-work terms between an outgoing spherical wavefront and the auxiliary fields consisting of incident and reflected plane waves. This integral can be evaluated by the stationary phase approximation for the two-dimensional case, or by a generalisation of this approximation for the three-dimensional case. Although this evaluation involves several distinct contributions, the final result is shown to be very simple, and it can be interpreted as a generalisation of a known result for the one-dimensional case, whereby the net contribution arises only from counter-propagating waves of the same mode. The results derived for a buried force are extended to the case of buried cracks by exploiting the body force equivalents for displacement discontinuities across a surface.