The mathematical formulation and evaluation of the radiation field for general elastodynamic sources is given and applications of the theory to the description of source fields of geophysical interest are treated. The study was primarily undertaken to provide a theoretical basis for estimating the properties of the tectonic stress field and parameters of rupture phenomenon in the earth through observations of the radiation field from earthquakes and other tectonic sources. Thus the description of the tectonic source is particularly emphasized, both as to its physical origins and with respect to the radiation field to be expected from it. The mathematical description of the tectonic source field is achieved in terms of an elastic relaxation theory of radiation which corresponds to a generalized initial value problem involving the initial prestress field. As a consequence, the radiation field is obtained in terms of the rupture expansion rate (velocity of rupture), the rupture dimensions and orientation and the magnitude and orientation of the initial stress field. Inertial conditions are inherent in the relaxation theory so that the time dependence of the field is automatically specified. Careful attention is given to causality relationships so that the resulting field expressions contain the complicated space-time relationships associated with a tectonic source field. Energy and equilibrium relations are considered and expressions are obtained for the estimated energy release and the final static field in terms of the source parameters. Detailed properties of the radiation field are given in the form of source field amplitude and phase spectra. Spatial radiation patterns are obtained showing the direction properties as functions of frequency, prestress and other source parameters. Similar results are given for shock induced rupture under prestress conditions, along with estimates of tectonic energy release. It is concluded that the theoretical predictions for the properties of the radiation field from a spontaneous rupture source are in general agreement with the actual observations of the field from such a source, but that accurate estimates of the prestress and rupture parameters require a more complete coverage and analysis of the field than is usually the case. It is concluded from a preliminary analysis of the Ranier nuclear explosion that tectonic energy release did occur and that the anomalous radiation observed would correspond to a prestress shear field of the order of 20 bars. The most likely mechanism of rupture at depth in the earth is considered to be unstable creep phenomenon resulting in phase change (melting) and the rupture source models adopted are not inconsistent with this hypothesis.