An analytical representation of the velocity potential for diffraction radiation of regular (time-harmonic) water waves by a 3D offshore structure is given. This flow representation is based on a weakly-singular boundary-integral potential-flow representation and a Green function that satisfies the radiation condition and the free-surface boundary condition in the farfield. This analytical flow representation provides an equation suited for determining the boundary potential, and a Galerkin solution-procedure is proposed.
Well-established panel methods for evaluating diffraction-radiation of regular (time-harmonic) water waves by 3D offshore structures are widely available, and routinely used in offshore hydrodynamics; e.g. Chen (2004). These methods are based on numerical solution of an integral equation, formulated using a Green function (a pulsating source below a free surface) in accordance with Green's classical approach, for the velocity potential at the mean wetted hull of the offshore structure. A variation on this classical "Green-function method" is presented here. The approach that is used in this study is based on a weakly-singular boundary-integral representation of the potential and a "farfield free surface Green function". This Green function satisfies the radiation condition, and also satisfies the boundary condition at the free surface in the farfield. In the nearfield, the free-surface boundary condition is only satisfied to leading order by the simple Green function employed here. The approach yields an analytical representation of the velocity potential at flow-field point within the mean flow domain, including the boundary surface (mean wetted hull of the offshore structure). This flow representation only involves ordinary (algebraic, trigonometric, and hyperbolic) functions real arguments, and it defines potential that is continuous at the boundary surface (unlike Green's classical potential representation, which involves a dipole distribution).
Nondimensional coordinates, flow velocity, and velocity potential are defined in terms of a reference length L, velocity U, and potential UL.
