Boundary Integral Methods for Three-Dimensional Water Waves

We describe a boundary integral method for computing time-dependent, doubly periodic, three-dimensional water waves. This method has been proved to converge to the exact solution. We discuss several analytical issues, including the quadrature of singular integrals, such as single and double layer potentials on surfaces, and stability estimates for discrete versions of operators such as the layer potentials. Water waves have long been studied because of their practical importance and because they ooer a familiar setting for a variety of phenomena of nonlinear wave motion. They have attracted the attention of mathematicians starting with Cauchy, Poisson, and Stokes. The full equations are diicult to deal with analytically, but there have been many signiicant applications of approximate models, primarily the linear theory and the shallow water theory. For example, Kelvin explained the wake pattern behind a ship at sea or a duck in a pond within the linear model using the method of stationary phase. The shallow water theory can describe motions such as the breaking of a dam; it is also used to model the atmosphere. Strong connections have been found between the behavior of water waves and integrable systems, such as the Korteweg-de Vries equation. General references include 8, 15, 17, 24, 29]. Understanding of the full equations can be sought through analysis and numerics. Existence results for the exact, time-dependent problem have been diicult to obtain, because of the free boundary and the essential nonlinearity. With no viscosity assumed, the motion has the character of a nonlocal, nonlinear wave equation. The rst existence result with a nite degree of smoothness was proved by Nalimov; his method was extended by Yosihara. More deenitive results were proved recently by Sijue Wu 30, 31]. She shows that the initial value problem in two or three space dimensions, without surface tension, has a smooth solution, locally in time. Surprisingly, the initial state may be one where the wave overturns, i.e., there is water above air. Numerical computations have often been used to simulate water waves in applied mathematics and in ocean engineering. Extensive codes have been developed to deal with practical problems such as the load on ooshore structures and the action of waves on ships