Nonlinear resonance of trapped waves on a plane beach

Abstract

Linear edge waves were first found mathematically by Stokes (1846). It has long been a topic of interest, since edge waves are believed to be responsible for the formation of beach cusps. Galvin (1965) was the first to observe in the laboratory that edge wave mode of frequency w can be excited by incident waves of frequency w. Guza and Bowen (1976) gave a theoretical explanation of the nonlinear mechanism of sub-harmonic resonance. This type of theory has been extended by Minzoni & Whitham (1977), Rockliff (1978), and Rockliff & Smith (1985), among others. Rockliff also initiated a. theory whereby subharmonic resonance can be achieved by an incident wave of frequency w, which leads to second harmonic at the second order to excite the edge wave. Their theory was however incomplete. The goal of this thesis is to extend the existing theories in order to show other paths to resonance. The nearshore region is a plane beach of small slope. For all cases we shall derive the evolution equations governing the edge wave amplitudes and analyze the stability of the equilibrium (static or dynamic) state. We first study in Chapter 2 synchronous resonance as a special case. We shall show that interaction between the edge wave and the incident/reflected wave also generates circulation cells on the beach.

(cont.) Comparison is made between our theoretical results and the experiments by Bowen & Inman. We then give a corrected version of the work of Rockliff and Rockliff & Smith. Next, in Chapter 3 we generalize the idea of Chapter 2 and examine the excitation of an edge wave by a pair of incident waves of magnitudes comparable to the saturated edge wave but the sum of their frequencies can cause subharmonic resonance nonlinearly. In the development of theory, singularities were discovered at certain incident wave frequencies, each of which coincides with the difference of natural frequencies of two edge wave modes. Hence lower-order resonance happens. To remove this singularity we consider in Chapter 4 the simultaneous excitation of two modes of edge waves by one incident wave. The two edge waves are shown to be coupled by both linear and nonlinear terms. Instead of a fixed point, the dynamical system has two limit cycles as dynamic equilibria. Nonlinear numerical simulation and linear instability are analyzed.