On water waves produced by ground motions

Abstract

A linear and irrotational model is constructed to represent the formation of water waves by ground motions of a sloping bed. A survey of the constant depth case, given first, helps in understanding the mechanism of formation, and, in this oversimplified case, wave propagation away from a source, which is usually very asymmetric. The importance of asymmetry, which may produce trapped waves, is illustrated by an estimate of the propagation in a three-dimensional case. The formation of waves by a ground motion on a slope is then studied in detail. The problem is reduced to linear integral equations of the first kind. Using an inversion technique one constructs a source–response pair in which the source is ‘δ-like’ and the response is close to that which would be found if the depth was constant around the source. A general approximate solution is then derived, in both the two-dimensional and three-dimensional cases. Results for the sloping-bottom case are given for small times. They give initial values of surface displacement. They also enable one to determine the important physical parameters in the ground motion and to evaluate the efficiency of wave production.