Nonlinear Dynamics of Deep-Water Gravity Waves

Abstract

This chapter reviews some recent progress in the nonlinear dynamics of deep-water gravity waves. It attempts to highlight the major developments in theory and experiment commencing with the finding by Lighthill (1965) that a nonlinear, deep-water gravity wave train is unstable to modulational perturbation, up to the present investigations of various aspects of nonlinear phenomena, including three-dimensional instabilities, bifurcations into new steady solutions, statistical properties of random wave fields, and chaotic behavior in time evolution. The governing equations for inviscid, irrotational, incompressible, free surface flows are given in Section II of the chapter, together with some basic steady solutions of the system. The concept of a wave train is introduced in Section III. The stability and evolutionary properties of a weakly nonlinear wave train in two dimensions are considered in Section IV, based on the nonlinear Schrodinger equation, which is an equation describing the wave envelope. Some interesting phenomena, such as the existence of envelope solitons, and the Fermi-Pasta-Ulam recurrence in time of an unstable wave train, are examined. Section V extends these results to three dimensions, using the three-dimensional nonlinear Schrodinger equation. The results indicate that whereas the nonlinear Schrodinger equation is remarkably successful in describing the two-dimensional dynamics, it is inadequate for treatment of the three-dimensional case.