A uniform asymptotic expansion for the shear-wave front in a layer

Abstract

In studying transient waves in a layer, a Fourier-type integral arises. For a material point at a position in a neighborhood behind the shear-wave front, the phase function of this integral has a stationary point which approaches positive infinity. As a result, the classical method of stationary phase does not apply. A heuristic treatment has been suggested by Jones ( Q. J. Mech. Appl. Math. 17, 401–421 (1964)), but it seems to give incorrect results. In this paper, an asymptotic expansion is derived for this integral, which is uniformly valid in a neighborhood behind the shear-wave front. Our result shows that there is a jump in the asymptotic order of the vertical acceleration behind and ahead of the shear-wave front. It is also found that there is a transition from an order O(1) disturbance to an order O(t − 1 4 ) disturbance as the distance to the shear-wave front increases.