Seismic radiation from a unidirectional asymmetrical circular crack model, part II: Variable rupture velocity

The unidirectional asymmetrical circular crack model proposed by Dong and Papageorgiou (2002) in a companion article (referred to in this article as Part 1) is further developed to account for variable rupture velocity using the analysis method proposed by Sato (1994). Special emphasis is placed on the properties of the radiated acceleration pulses emitted by a rapid but continuous change in rupture velocity. The mathematical expressions describing such pulses are valid even if only a segment of the rupture front (in the vicinity of the critical point/zone that is responsible for their emission) is coherent. Furthermore, they are valid also in the near field. All the conclusions reached by Sato (1994) for the symmetrical circular crack model, appropriately modified to account for azimuthal variation of directivity effects, are valid also for the unidirectional asymmetrical circular crack model. Specifically, the acceleration pulse exhibits a directivity with respect to both the pulse width and amplitude. The pulse width is, on average, given by the directivity factor, which depends on the rupture velocity averaged over the time interval during which the change of rupture velocity takes place. Furthermore, the directivity factor, as expected, demonstrates a strong azimuthal dependence with the strongest radiation channelled in the direction of crack growth. The pulse width radiated toward the growth direction of the rupture front is shorter than that radiated toward the opposite side. The spectral amplitude of the acceleration pulse depends linearly on the strain drop over the crack surface, the radius of curvature of the rupture front at which the rupture velocity starts to change, the magnitude of the change in rupture velocity, and the generalized radiation pattern coefficient. The directivity of the radiation pattern coefficient is stronger than that of the case of an abrupt change in rupture velocity.