A Nonlocal Formulation of Water Waves

A Nonlocal Formulation of Water Waves

Terry Haut
Department of Applied Mathematics
University of Colorado at Boulder

The classic water wave problem consists of finding the evolution of an ideal fluid with a free interface. Because of the free interface, the classic water wave equations are highly nonlinear, and are therefore difficult to analyze theoretically and numerically. We discuss a recent reformulation of water waves, given in terms of a new nonlocal equation that connects the free interface and the velocity potential evaluated on the free interface. This integral equation, together with the well-known Bernoulli equation (expressed in terms of the interface variables), serve to replace the standard water wave equations. One notable advantage of this reformulation is that the vertical coordinate is eliminated, thus reducing the dimensionality of the problem and fixing the domain in which the equations are posed.