On the shore singularity of water–wave theory. II. Small waves do not break on gentle beaches

Abstract

The model of gravitational surface waves on beaches of small slope formulated in Part I [Phys. Fluids 2 9, ▪▪▪ (1986)] and its mathematical theory [R. E. Meyer, Adv. Appl. Math. (in press)] are used to show how an incident‐wave amplitude can be defined so that a bound on it guarantees solutions which respect the assumptions of the model everywhere and forever. The structure of those solutions ‘‘far’’ from shore is then compared with that predicted ‘‘near’’ shore by the classical, linear theory [Commun. Pure Appl. Math. 1 (1948)] to remove the indeterminacies of both theories and to develop a unified theory which describes the whole shoaling process for unbroken waves of arbitrary time‐dependence on inviscid water. These results indicate that the beach theory [Waves on Beaches (Academic, New York, 1972), p. 357 and Phys. Fluids 2 9, ▪▪▪ (1986)] captures and elucidates the basic singularity structure underlying the shore behavior of gravitational surface waves.