An investigation into translating pulsating Green' functions for ship motion prediction

During recent years increasing interest has been shown in the theoretical calculation of three dimensional ship motions using singularity distribution methods and the coupled element method. Any practical calculation, particularly for a surface ship requires an efficient numerical scheme for evaluating the fundamental Green's function. This Green's function has been formulated as a double integral, most of the recent work has been to define alternative single integral forms making use of the complex exponential integral. In this paper the Green's function derived by Wu & Eatock Taylor is used. A major difficulty in its evaluation is the presence of a singularity in the integrand. A method is described, whereby this singularity is subtracted out to the first order and evaluated analytically. The Green's function is then evaluated analytically, and further calculated using an adaptive trapezium rule and Romberg quadrative. The behaviour of the Green's function for various frequencies, speeds and singularity positions typically used in ship motion predictions are shown and the implications discussed.