||Recent developments in acoustic and electromagnetic diffraction theory show that the formulation of diffraction problems in terms of integral equations is a subject of growing importance (see Bouwkamp (13)). Therefore, it seems worth while to attempt a generalization of the relevant methods to the field of elasto-dynamic diffraction theory. Now it is a well-known fact that in a homogeneous, isotropic, elastic solid there are two velocities of propagation; the larger of the two is associated with the wave fronts of irrotational or compressional waves, the smaller of the two is associated with the wave fronts of equivoluminal or shear waves. In a medium of infinite extent the two types of waves can propagate independently; however, as soon as boundaries occur, an interaction between the two types of waves takes place. Therefore, the phenomena related to the diffraction of elastic waves are expected to be of a complicated nature.
One of the most important applications of the theory of elastic wave propagation is the field of seismology. This explains why the emphasis is not on the steady-state behaviour of a system but rather on its transient response to a source which starts to act at a certain instant. Also, most of the problems that have been investigated deal with the radiation from a source located in an elastic medium consisting of several layers with different elastic properties (model of the earth). In this respect we mention Lamb's (26) classical solution of the problem of the radiation from a line source or a point source located at the free surface bounding an elastic half-space. A recent book by Ewing, Jardetzky and Press (16) covers most of the work that has been done on this type of problems.
Another publication we want to mention is Cagniard's monograph (14) on the generalization of Lamb's problem to the case of a point source located in one of two coupled elastic half-spaces. In this monograph the author develops a general method of solving transient problems. The idea is roughly as follows. After having taken the Laplace transform with respect to time, the remaining boundary value problem is solved. The solution of this boundary value problem is then written in such a form that the transient problem under consideration can be solved more or less by inspection and not by evaluating a Mellin inversion integral.