A simplified derivation of the ordinary differential equations for the free-surface Green functions

Ordinary differential equations (ODEs) are derived for the free-surface Green functions and their gradients, in a fluid of infinite depth. The cases of harmonic time-dependence in the frequency domain and impulsive motion in the time domain are considered separately. These ODEs were derived originally by Clément, starting with fourth-oder equations in the time domain and using Fourier transforms to derive a second-order ODE in the frequency domain. In the present work a simpler procedure is followed independently in each domain, transforming the governing Laplace equation to an ordinary differential equation. The results are consistent with the ODEs derived using Clément’s method.