We analyze three-dimensional finite difference solutions for a simple shear-crack model of faulting to determine the effects of fault length and width on the earthquake slip function. The fault model is dynamic, with only rupture velocity, fault dimensions, and dynamic stress-drop prescribed. The numerical solutions are accurate for frequencies up to 5 Hz, and are combined with asymptotic results for shear cracks in order to characterize the slip function at higher frequencies.
Near the hypocenter, the slip velocity exhibits a square-root singularity whose intensity increases with hypocentral distance. At distances greater than the fault width, w, growth of the velocity intensity ceases, and the slip function becomes nearly invariant with distance along the fault length. Closed-form expressions are developed for the dependence of static slip (s∞), slip rise time (TR), and slip velocity intensity (V) on fault geometry. Along the center line of a long, narrow fault, at hypocentral distances exceeding w, these expressions reduce to s∞ ≈ wΔτ/μ, TR ≈ 0.5 w/vR, and V ≈ √w/2 vR Δτ/μ, where Δτ is the dynamic stress drop, μ the shear modulus, and vR the rupture velocity.
The numerical results imply that uniform-dislocation kinematic earthquake models in which slip is represented by a ramp time function will underpredict high-frequency ground motion relative to low-frequency ground motion. A further implication of the numerical solutions is that the nature of inelastic processes at the advancing edge of a long fault will depend on fault width, but will be independent of rupture length.