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 for ms are designed such that they have minimal wave resistance in calm wa ter. 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. Fi rst, the appropriate free-surface condition is shown. Then, the standa rd ray method, well-known in geometric optics, is formulated in the fl uid region and at the free surface. After an elimination process the e ikonal equation and the transport equation are obtained. The character istic equations for the nonlinear eikonal equations are derived, keepi ng in mind that the characteristics (rays) are not perpendicular to th e wave fronts, due to the influence of the double-body potential gener ated by the slow forward speed of the ship, which is assumed to be a g ood approximation for the steady potential. Numerical integration of t he ray equations lead to the ray pattern. After some manipulations the amplitude may be computed just as well. Finally, the second order mea n force or added resistance is calculated. The pictures of the ray pat terns show a caustic for values of the dimensionless parameter tau = o mega U/g > 1/4, where omega is the frequency of the incident wave with respect to the ship and U is the speed of the ship. To analyse the be haviour of the surface elevation near the caustic we consider the two- dimensional problem of the diffraction of short waves by a two-dimensi onal cylinder in a current U. Near the point where the local value tau = omega u(r)/g = 1/4 a boundary layer expansion leads to a uniformly valid expansion in terms of Airy functions, as in geometric optics. T he singular behaviour of the outer solution near this point can be eva luated numerically by integration of the eikonal equation. The final m atching is carried out analytically.