A new formulation of the Green function in water of finite depth at low frequencies

Abstract

We consider the frequency-domain free-surface Green function (GF) of wave radiation and diffraction in water of finite depth which is important for the hydrodynamic analysis of floating bodies, in general, and some renewable devices in shallow water, in particular. Further to the sound decomposition formulation presented in Chen (1993), in which two depth-effect functions are defined, a new analysis has been performed to obtain a new formulation. It is composed of the analytical part including all singular behaviors and oscillations, and the remaining part is well suited for numerical computations with an accuracy of order 10−8. Furthermore, the formulation is developed that the remaining part is fully smoothly varying and then approximated by Chebyshev polynomials. Not only the Green function itself but also its first-order and second-order derivatives are numerically evaluated and approximated independently by polynomials. The analytical formulations plus polynomials are then used to enhance efficiency and accuracy. Unlike previous studies, the analytical part of the Green function contains explicitly the logarithmic singularity when the depth-scaled wavenumber tends to zero. Classically included in the ’smooth’ part, this singular term is then extracted and the remaining part is fully smooth and much better approximated with consistent accuracy of 10−6 and less partitions. Furthermore, the added-mass and damping coefficients around a hemisphere are given.