SEISMIC WAVES IN TRANSITION LAYERS*

Abstract

A bstract A transition layer is a layer in which the velocity changes linearly with depth from v a to v e and we consider it embedded between two layers of constant velocity v a and v e , respectively. A plane displacement wave of arbitrary shape is supposed to come out of the upper v”‐layer, and to meet the transition layer at vertical incidence. Using methods of elementary calculus only, the subsequent events are derived for both inside and outside the transition layer. The transition layer can be regarded as the limit of a sequence of constant‐velocity‐layer divisions. Thus, after computing the first few multiples of the incident wave for these divisions, the respective multiples of the transition layer are obtained as the limit of these multiples. Then, with the help of recurrence formulae, the multiples of the transition layer of any order are computed. The sum of all multiples defines the complete response of the transition layer and satisfies the differential equations of the problem. The solution has the form of a superposition integral, and it is seen that the transition layer has the properties of a linear filter. The superposition integral is built up out of the incident wave and a conglomerate of Bessel functions, the latter being the corresponding response to an incident spike impulse in a mathematical sense. The reflected and transmitted responses to the spike impulse are shown for two values of the velocity ratio. For large values of this ratio, both these responses have a‘wave‐like’shape and it is seen that the transition layer may effect a serious change of shape of incident waves.