Total energy and energy spectral density of elastic wave radiation from propagating faults

Starting with a Green's function representation of the solution of the elastic field equations for the case of a prescribed displacement discontinuity on a fault surface, it is shown that a shear fault (relative displacement parallel to the fault plane) is rigorously equivalent to a distribution of double-couple point sources over the fault plane. In the case of a tensile fault (relative displacement normal to the fault plane) the equivalent point source distribution is composed of force dipoles normal to the fault plane with a superimposed purely compressional component. Assuming that the fault break propagates in one direction along the long axis of the fault plane and that the relative displacement at a given point has the form of a ramp time function of finite duration, T, the total radiated P and S wave energies and the total energy spectral densities are evaluated in closed form in terms of the fault plane dimensions, final fault displacement, the time constant T, and the fault propagation velocity. Using fault parameters derived principally from the work of Ben-Menahem and Toksöz on the Kamchatka earthquake of November 4, 1952, the calculated total energy appears to be somewhat low and the calculated energy spectrum appears to be deficient at short periods. It is suggested that these discrepancies are due to over-simplification of the assumed model, and that they may be corrected by (1) assuming a somewhat roughened ramp for the fault displacement time function to correspond to a stick-slip type of motion, and (2) assuming that the short period components of the fault displacement wave are coherent only over distances considerably smaller than the total fault length.