Slowly moving hull forms in short waves

The purpose of this paper is to provide a mathematical tool to improve the optimal design of ship forms. It is common practice that hull forms are designed such that they have minimum wave resistance in calm water. In this paper a theory is described by which the effect of short waves may be incorporated.The basic tool we use is the ray theory. First, the appropriate free-surface condition is shown. Then, the standard ray method, well-known in geometric optics, is formulated in the fluid region and at the free surface. After an elimination process the eiconal equation and the transport equation are obtained. The characteristic equation for the nonlinear eiconal equation is derived, keeping in mind that the characteristics are not perpendicular to the wave fronts, due to the effect of the double-body potential due to the forward speed of the ship, which is assumed to be a good approximation for the steady potential.Numerical computations are carried out by means of the RK4 method to obtain the ray pattern. After some manipulations the amplitude may be computed just as well. Finally, the nonlinear added-resistance force is calculated. Pictures of ray patterns for several angles of incidence are shown. Also the forces are shown.