Variational principle for determining the unknown wetted surface of a planing ship

A variational principle is presented to solve the problem of determining the unknown wetted surface of a planing ship on still water under gravity. The functional associated with the variational principle is expressed by the unknown wetted length distribution, the vortex line function, and the circulation distribution around the longitudinal sections, not by the bottom pressure distribution. In addition, the variational principle is adjoined with pseudoflow, here called reverse flow. The extremal of the functional satisfy the hull boundary integral equation in a similar way to that of lifting-surface theory and the elevated water surface condition along the spray root line. Both are sufficient to determine the wetted surface. A method of high aspect ratio approximation is also investigated. A simple relation is obtained between the wetted surface when moving and at standstill. The calculated shapes of the wetted surface of a planing plate agreed well with experimental results. Received for publication on Nov. 11, 1999; accepted on April 4, 2000