On the shore singularity of water waves. I. The local model

Abstract

The nonlinear equations governing inviscid water waves close to shore over beaches of small slope are discussed to elucidate their scope, limitations, and ramifications. Their decisive feature is a singularity of the differential equations themselves, and to open the way for analysis, an associated problem is formulated in terms of singular wave equations. The emphasis is on the identification of the questions of oceanographical relevance that can be asked and the ways in which they can be posed to construct the basis for a correct, mathematical analysis [R. E. Meyer, Adv. Appl. Math. (in press)] and for its application to shoaling predictions [R. E. Meyer, Phys. Fluids 2 9, XXX (1986)].