The Green function in the theory of radiation and diffraction of regular water waves by a body

Abstract

This study is concerned with the Green function of the theory of potential flow about a body in regular (time-harmonic) water waves in deep water, that is with the linearized velocity potential of the flow due to a source of pulsating strength at a fixed position below the free surface (or a pulsating flux across the free surface) of a quiescent infinitely deep sea. An asymptotic expansion and a convergent ascending-series expansion for the Green function are obtained from two alternative complementary ‘near-field’ and ‘far-field’ single-integral representations in terms of the exponential integral. The asymptotic expansion and the ascending series allow efficient numerical evaluation of the Green function for large and small distances, respectively, from the mirror image of the singularity (submerged source or free-surface flux) with respect to the mean sea surface.