Abstract An integral solution to the differential equations of elastodynamics is given. The solution for the displacement at any point in an isotropic homogeneous, elastic medium is obtained in terms of the initial distribution ot body forces within a given volume and the initial distribution of displacements and stresses over the bounding surface of the volume. These quantities appear in the solutions as retarded displacements, stresses, and body forces, with two types of retardation depending upon the velocities of longitudinal and transverse waves. From the integral solution the diffraction of elastic waves through large apertures bounded by opaque walls in solid materials is formulated. The incident wave motions may be either of longitudinal or transverse polarizations. The expected shear-compression interaction is obtained in the course of the computation. By analogy with optical procedure, Fraunhofer diffraction of elastic waves may be investigated as a special case. In the example of diffraction by a slit, an incident compression wave is diffracted into (1) a compression wave with the “ordinary” spatial distribution similar to that obtained in the corresponding optical problem and (2) a weaker shear wave with an “extraordinary” distribution. In the case of incident shear waves, there is obtained a diffracted shear wave with an ordinary distribution, and a possible weak compression wave with the extraordinary distribution. The presence of the diffracted compression wave depends Upon the polarization of the incident shear wave. Opacity in the Kirchhoff sense is a property of materials which are perfectly rigid.

Abstract The wide acceptance of the plate-tectonics model to describe inferred large-scale motions of the earth at the surface has prompted considerable activity in attempts to construct models of circulation of the mantle. These models must be consistent not only with the observations of surface motions, which are kinematic constraints, but also with certain rheological and geophysical constraints. Additionally, the driving mechanism must be sufficiently potent to overcome the energy losses in the system which are, minimally, the friction in the circulation as a result of earthquakes and viscous-drag forces. To date, only processes of thermally driven convection seem to be energetically potent. The source of the heat has been variously proposed to be one or more of the following: the core, radioactivity in the mantle, differentiation of mantle and crustal material, and differentiation of radioactivity into the crust. Deceleration of the earth's rotation is only marginally potent enough to generate plate motions. The choice of rheologies which have been proposed ranges from those for materials with finite strength, and with properties of brittle fracture, to those which exhibit constant viscosities under linear stress-strain relations. Undoubtedly, both these extremes are inappropriate to the entire mantle. In the case of the first extreme, it has been shown that convective motions are not possible if the mantle rheology is similar to that of materials under-going brittle fracture at the surface of the earth. However, this rheology is inappropriate because the influence of increasing temperature and pressure weakens earth materials. A more appropriate rheology