New formulations of the ship-motion Green function

The ship-motion Green function representing the potential flow generated by a time-harmonically pulsating source with unit strength advancing beneath a free surface along a straight path at a constant speed is studied in the paper. Starting with the form of a double Fourier integral with respect to wavenumber k and polar angle \(\theta \) on the Fourier plane, one alternative formulation based on the single integral along dispersion curves for the wave component, and the integrals in polar angle after having evaluated integration in wavenumber, is first presented. Another new formulation based on performing the integral with respect to polar angle \(\theta \) prior to the wavenumber k is developed afterward. Analytical formulations of the \(\theta \)-integral defined as \(g_{\ell }\left( k\right) \) are obtained by applying the theorem of residue. The behaviours of poles located in the integration contour are analyzed, and a method to efficiently evaluate the integral of \(g_{\ell }\left( k\right) \) for different values of Brard number \(\tau \) is outlined. Furthermore, numerical calculation is implemented. On comparing with the results obtained by means of the existing formulation, an excellent agreement is achieved with a maximum of difference in the order of \(10^{-6}\) which shows that two formulations are intrinsically consistent. The new formulation based on the polar angle integration is shown to be suitable for the study of the spatial integration of the Green function.