Free-Surface Green Function
The Free-Surface Green function is a function which satisfies the following equation (in Finite Depth)
[math]\displaystyle{ \nabla_{\mathbf{y}}^{2}G(\mathbf{x},\mathbf{y})=\delta(\mathbf{x}-\mathbf{y}), \, -\infty\lt z\lt 0 }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z}=0, \, z=h, }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z} = k_{\infty}\phi,\,z\in\Gamma_s, }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z} = L\phi, \, z\in\Gamma_w. }[/math]
where [math]\displaystyle{ k_{\infty} }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ k_{\infty}=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity.
Many expressions for the Green function have been given. In two dimensions it can be written as
In three dimensions and infinite depth the Green function [math]\displaystyle{ G }[/math], for [math]\displaystyle{ r\gt 0 }[/math], was given by {\em Havelock\/} \cite{havelock55} as
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} \, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} - \frac{1}{\pi^2} \int\limits_{0}^{\infty} \frac{\alpha}{\eta^2 + \alpha^2} \big( \alpha \cos \eta (z+c) - \eta \sin \eta (z+c) \big) K_0(\eta r) d\eta, }[/math]
where [math]\displaystyle{ H^{(1)}_0 }[/math] and [math]\displaystyle{ K_0 }[/math] denote the Hankel function of the first kind and the modified Bessel function of the second kind, both of order zero as defined in {\em Abramowitz \& Stegun} \cite{abr_ste}. This Green function will be referred to as {\em Havelock\/}'s Green function. It should be noted that {\em Havelock\/}'s Green function can also be written in the following closely related form,
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} \, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty} \frac{(\eta^2 - \alpha^2) \cos \eta (z+c) + 2 \eta \alpha \sin \eta (z+c)}{\eta^2 + \alpha^2} K_0(\eta r) d\eta }[/math]
\cite{linton01}. An equivalent representation is due to {\em Kim} \cite{kim65} for [math]\displaystyle{ r\gt 0 }[/math], although implicitly given in the work of {\em Havelock} \cite{havelock55}, and is given by
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} - \frac{\alpha}{4} e^{\alpha (z+c)} \Big( &\mathbf{H}_0(\alpha r) + Y_0(\alpha r) - 2i J_0 (\alpha r) + \frac{2}{\pi} \int\limits_{z+c}^0 \frac{e^{-\alpha \eta}}{\sqrt{r^2 + \eta^2}} d\eta \Big), }[/math]
where [math]\displaystyle{ J_0 }[/math] and [math]\displaystyle{ Y_0 }[/math] are the Bessel functions of order zero of the first and second kind and [math]\displaystyle{ \mathbf{H}_0 }[/math] is the Struve function of order zero.
