Waves on the ocean's surface play important roles in weather forecasting and climate modeling, in the safety of coastal communities and offshore industries, and in overseas shipping. In this project, the investigators focus on physical effects that are often approximated or neglected altogether, but that are needed to predict accurately the observed behavior of ocean waves. Examples include the dissipation of ocean swell during propagation across the deep ocean and subsequently onto the shoreline; the time-dependence of wind in the wave-generation process; and dispersion of waves in shallow water. The inclusion of dissipation will lead to a better understanding of how wave energy evolves. The inclusion of time-dependence in the wind that generates waves will allow for a better understanding of the initial period during which energy is transferred from air to water. The inclusion of dissipation and dispersion in models for shallow-water waves will allow for better predictive capabilities of waves in coastal areas. The research tools of the investigators include modeling, analysis, computer simulations, and laboratory experiments. While the emphasis is on water waves, for which they can conduct laboratory experiments, the mathematical analysis is more broadly applicable to other physical systems and is of interest in the study of partial differential equations.
The investigators propose analytic, numerical, and experimental investigations of the following: (A) Deep-water waves. They consider the frequency downshifting of freely propagating waves and wave generation due to wind. They are considering two models of frequency downshifting that differ in how the rotational part of the flow is modeled. To model wind-generated waves, they are allowing for time-dependent shear flows in both the air and water. The resulting stability problem for waves is non-standard, and understanding how to address it is a central mathematical question. (B) Shallow-water waves. They seek accurate models of dispersion and of dissipation due to the bottom, wall, and surface boundary layers. They will start with a Whitham equation and generalize it to include surface tension effects, dissipative effects, nonhorizontal bathymetry, and bidirectional waves. They will look, both analytically and numerically, for small- and large-amplitude solutions, and study their stability. They will further investigate how best to include dissipation that is due to the bottom boundary layer by comparing numerical simulations and experiments. (C) Three-wave partial differential equations. The three-wave partial differential equations, which arise in many physical applications, describe the simplest possible nonlinear interactions among dispersive wave trains, without dissipation. The investigators propose a solution method using a Painleve-analysis to obtain the general solution for arbitrary boundary conditions. There are few examples of general solutions of partial differential equaitons, so their adding one more example would be a mathematical breakthrough.
