Of strong practical interest in the subject of fracture mechanics and related structural design and damage are the surface-breaking and sub-surface cracks. Indeed the scattered-wave fields generated by the interaction of incident surface or body waves with these cracks would be expected to yield most of the important in-formation about the geometries of the cracks. It follows, in the subject of quantitative non-destructive evaluation (QNDE) there is considerable interest in scattering by surface-breaking and sub-surface cracks, as important steps in solving the inverse problems of obtaining the crack geometries from the scattered wave fields. However, having the solutions to the corresponding direct problems is a prerequisite to the inverse problems.