On the Mathematical Theory of the Motion of Floating Bodies - An Update.

Abstract : The generation of surface waves in a fluid caused by a partially submerged body is modeled mathematically as a boundary value problem for the Laplacian with Neumann data on the bottom of the fluid container and on the body, a linearized free surface condition and a radiation condition at infinity. Fritz John (Comm. Pure Appl. Math., III, 1950) showed that the problem had a unique solution only under rather restrictive assumptions on the body geometry and reformulated the problem as an integral equation which was not uniquely solvable at certain irregular frequencies. In the present work, uniqueness is established for more general geometries, allowing corners and nonnormal intersections of the body with the free surfaces. Also presented are two methods of modifying the integral equation so that it is uniquely solvable for all frequencies. One method involves introducing an additional integral on the waterplane, while in the second method, an additional integral term is added to the equation which remains an equation only over the submerged portion of the body.