SEISMIC WAVES DUE TO A SHEAR FAULT IN A SEMI-INFINITE MEDIUM

Abstract

In order to clarify basic characteristics of seismic waves in the near-field as well as in the far-field, exact solutions for free surface displacements generated from a shear fault with an arbitrary orientation in a semi-infinite medium are obtained in a cylindrical coordinate system. First, taking the free surface effects into account, expressions for Laplace transforms of displacements with respect to time are derived, and secondly exact transient solutions are obtained by using the Cagniard's method which gives the inverse Laplace transforms in a very ingenious manner when the source time function is of the ramp type. In sections 2, 3 and 4, mathematical expressions are derived, and the results and interpretations of numerical computations for a point source are presented in section 5. Basic characteristics of each phase are summarized as follows: P pulse has basically a rectangular form. The initial pulse amplitudes in a semi-infinite medium are, even in rather near-field, close to those in an infinite medium with correction of surface effects due to plane wave incidence. SP pulse, which radiates from the source as S phase, is incident onto the free surface with a critical angle and then is propagated along the surface with the speed of P-wave velocity, has a relatively large amplitude in the near-field and cannot be neglected when the wave form on the free surface is discussed. This pulse is observed when the epicentral distance is greater than the critical distance. S pulse forms are quite different at epicentral distances less than and greater than the critical distance. S pulse beyond the critical distance has logarithmic infinities at the arrival time of S phase, tS, and tS+t0, t0 being the rise time of the source function. Therefore a plane-wave correction can-not be applied successfully as in the case of an onset of the P pulse. The Rayleigh pulse is well developed when the epicentral distance is about five to ten times as large as the focal depth and its form is not very much affected by the rise time of the source function. For surface focus, S pulse has no logarithmic infinity but the Rayleigh pulse has infinities at the arrival time tR and tR+t0. It is shown that the solutions for a moving source can be obtained by numerical integrations of the solutions for the point source. This case will be dealt with in a subsequent paper.