The effect of the rupture velocity upon the spatial and temporal dependence of earthquake source functions is investigated. To this end a model is suggested in which the fault zone is realized as a flexible membrane under the action of a moving force with additional stiffness forces provided by the surrounding medium. The motion of each particle of the membrane is impeded by a displacement-dependent friction and radiation damping.

The particle motion along the fault is found to obey an inhomogeneous Klein-Gordon equation whose solutions are derived in closed form. In the time domain, the solutions yield a particle-motion function that has frequently been derived by analysis of earthquake seismograms. The physical parameters in the theoretical source function are found to depend strongly on the Mach number, as already predicted by the theoretical directivity function.

The theory excludes the possibility of supersonic rupture and asserts a transonic rupture for major shallow earthquakes and subsonic rupture for seismic events with low and intermediate magnitudes. It predicts new functional relations between the initial particle velocity at the fault's tip, 0, the Mach number, M = ν/β, the rise time τ, the stress drop σ and the fault length L.

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