Theories of seismic source radiation differ, in their predictions of the long-period spectrum. These theories are examined in terms of the assumptions made in the models used and it is concluded that, despite the simple source geometry used, the elastodynamic relaxation theory due to Archambeau is physically more realistic than theories based on either cavity models or propagating point sources. A study of errors arising in frequency analysis of standard seismic records reveals that it is difficult to determine accurately the amplitude spectrum at long periods. Provided requirements of sample length and signal-to-noise ratio are met, displacement spectra of body waves are characterised by dominant periods as predicted by relaxation theory. A modification of Archambeau's spherical model is proposed such that strain energy is released from a medium as a result of slip across a thin molten disk. If the energy loss is that required to melt the disk then radiation efficiency increases with event size. Numerical solutions of theoretical spectra for a wide range of parameters shows that rupture dimensions can be calculated from observed dominant periods. Analysis of a set of South American deep focus events (600 km depth) in the magnitude range 3.8 to 6.1 yields P wave dominant periods between 1.5 sand 30 s, and hence rupture dimensions varying from 3 km to 60 km. This is consistent with dimension estimates made on the assumption that the duration of high frequency radiation is a measure of the geometric size of an event. Radiated energies are calculated by integrating over the arrivals and correcting back to the sources. For all but the largest events, the dominant periods vary with radiated energy as predicted by the source model. Interpretation in terms of the model yields a stress drop (assumed complete) of 3 to 5 bars and a radiation efficiency varying from about 0.1 for magnitude 4 events to 0.8 or 0.9 for very large events. Because the fault plane area/energy ratios are high, simple localised melting is precluded as the source mechanism. Results from a set of events at 200 km depth are inconclusive because of inadequacies in the location data. Provided these difficulties are resolved, extension of this investigation should provide an estimate of the variation of seismic stress with depth.