Explicit representation for the infinite-depth two-dimensional free-surface Green’s function in linear water-wave theory

Abstract

In this paper we derive an explicit representation for the two-dimensional free-surface Green’s function in water of infinite depth, based on a finite combination of complex-valued exponential integrals and elementary functions. This representation can easily and accurately be evaluated in a numerical manner, and its main advantage over other representations lies in its simplicity and in the fact that it can be extended towards the complementary half-plane in a straightforward manner. It seems that this extension has not been studied rigourously before and it is required when boundary integral equations are extended in the same way. For the computation of the Green’s function, the limiting absorption principle and a partial Fourier transform along the free surface are used. Some of its properties are also discussed, and an expression for its far field is developed, which allows to state appropriately the involved radiation condition. This Green’s function is then used to solve the two-dimensional infinite-depth water-wave problem by developing a corresponding boundary integral equation, whose solution is determined by means of the boundary element method. To validate the computations, a benchmark problem based on a half-circle is presented and solved numerically.