The diffraction of short free-surface water waves, a uniform expansion

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 minimal 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 eikonal equation and the transport equation are obtained. The characteristic equations for the nonlinear eikonal equations are derived, keeping in mind that the characteristics (rays) are not perpendicular to the wave fronts, due to the influence of the double-body potential generated by the slow forward speed of the ship, which is assumed to be a good approximation for the steady potential. Numerical integration of the ray equations lead to the ray pattern. After some manipulations the amplitude may be computed just as well. Finally, the second order mean force or added resistance is calculated. The pictures of the ray patterns show a caustic for values of the dimensionless parameter τ = ωU g > 1 4 , where ω is the frequency of the incident wave with respect to the ship and U is the speed of the ship. To analyse the behaviour of the surface elevation near the caustic we consider the two-dimensional problem of the diffraction of short waves by a two-dimensional cylinder in a current U. Near the point where the local value τ ∗ = ωU r g = 1 4 a boundary layer expansion leads to a uniformly valid expansion in terms of Airy functions, as in geometric optics. The singular behaviour of the outer solution near this point can be evaluated numerically by integration of the eikonal equation. The final matching is carried out analytically.