Free-Surface Green Function
Introduction
The Free-Surface Green function is one of the most important objects in linear water wave theory. It forms the basis on many of the numerical solutions, especially for bodies of arbitrary geometry. It first appeared in John 1949 and John 1950.
There are many different representations for the Green function.
Equations for the Green function
The Free-Surface Green function is a function which satisfies the following equation (in Finite Depth)
[math]\displaystyle{ \nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\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=0. }[/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. We also require a condition as [math]\displaystyle{ \mathbf{x} \to \infty }[/math] which is the Sommerfeld Radiation Condition. This depends on whether we assume that the solution is proportional to [math]\displaystyle{ \exp(i\omega t) }[/math] or [math]\displaystyle{ \exp(-i\omega t) }[/math]
We define [math]\displaystyle{ \mathbf{x}=(x,y,z) }[/math] and [math]\displaystyle{ \mathbf{\xi}=(a,b,c) }[/math]
Two Dimensional Representations
Many expressions for the Green function have been given. We present here a derivation for finite depth based on an Eigenfunction Matching Method. We write the Green function as
[math]\displaystyle{ G(x) = \sum_{n=0}^\infty a_n(x)f_n(z) }[/math]
where
[math]\displaystyle{ f_n(z)=\frac{\cos(k_n(z+h))}{N_n} }[/math]
[math]\displaystyle{ k_n }[/math] are the roots of the Dispersion Relation for a Free Surface
with [math]\displaystyle{ k_0 }[/math] being purely imaginary with negative imaginary part and [math]\displaystyle{ k_n, }[/math] [math]\displaystyle{ n\geq 1 }[/math] are purely real with positive real part ordered with increasing size. [math]\displaystyle{ N_n }[/math] is chosen so that the eigenfunctions are orthonormal, i.e.,
The Green function as written needs to only satisfy the condition
We can expand the delta function as
Therefore we can derive the equation
so that we must solve
This has solution
The Green function can therefore be written as
[math]\displaystyle{ G(x) = \sum_{n=0}^\infty e^{|x-a|k_n}f_n(c)f_n(z) }[/math]
It can be written as
[math]\displaystyle{ -\sum_{n=0}^\infty \frac{ie^{i k_n}}{\tan(k_m H) + H k_m\sec(k_m)^2} }[/math]
Incident at an angle
In some situations the potential may have a simple [math]\displaystyle{ e^{i k_y y} }[/math] dependence and we require the Green function to satisfy the following equation
[math]\displaystyle{ \left(\partial_x^2 + \partial_z^2 +k_y^2\right) G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\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=0. }[/math]
The Green function be derived exactly as before expect we have to include [math]\displaystyle{ k_y }[/math]
[math]\displaystyle{ G(x) = \sum_{n=0}^\infty e^{|x-a|}\sqrt{k_n^2-k_y^2}f_n(c)f_n(z) }[/math]
Three Dimensional Representations
Let [math]\displaystyle{ (r,\theta) }[/math] be spherical coordinates such that
[math]\displaystyle{ x - a = r \cos \theta,\, }[/math]
[math]\displaystyle{ y - b = r \sin \theta,\, }[/math]
and let [math]\displaystyle{ R_0 }[/math] and [math]\displaystyle{ R_1 }[/math] denote the distance from the source point [math]\displaystyle{ \mathbf{\xi} = (a,b,c) }[/math] and the distance from the mirror source point [math]\displaystyle{ \bar{\mathbf{\xi}} = (a,b,-c) }[/math] respectively, [math]\displaystyle{ R_0^2 = (x-a)^2 + (y-b)^2 + (z-c)^2 }[/math] and [math]\displaystyle{ R_1^2 = (x-a)^2 + (y-b)^2 + (z+c)^2 }[/math].
Finite Depth
The most important representation of the finite depth free surface Green function is the eigenfunction expansion given by John 1950. He wrote the Green function in the following form
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{i}{2} \, \frac{\alpha^2-k^2}{(\alpha^2-k^2)H-\alpha}\, \cosh k(z+H)\, \cosh k(c+H) \, H_0^{(1)}(k r) + \frac{1}{\pi} \sum_{m=1}^{\infty} \frac{k_m^2+\alpha^2}{(k_m^2+\alpha^2)H-\alpha}\, \cos k_m(z+H)\, \cos k_m(c+H) \, K_0(k_m r), }[/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 Abramowitz and Stegun 1964, [math]\displaystyle{ k }[/math] is the positive real solution to the Dispersion Relation for a Free Surface and [math]\displaystyle{ k_m }[/math] are the imaginary parts of the solutions with positive imaginary part. This way of writing the equation was primarily to avoid complex values for the Bessel functions, however most computer packages will caculate Bessel functions for complex arguement so it makes more sense to write the Green function in the following form
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{1}{\pi} \sum_{m=0}^{\infty} \frac{k_m^2+\alpha^2}{(k_m^2+\alpha^2)H-\alpha}\, \cos k_m(z+H)\, \cos k_m(c+H) \, K_0(k_m r), }[/math]
where [math]\displaystyle{ k_m }[/math] are as before except [math]\displaystyle{ k_0=ik }[/math].
An expression where both variables are given in cylindrical polar coordinates is the following
given by Black 1975 and Fenton 1978
Infinite Depth
In three dimensions and infinite depth the Green function [math]\displaystyle{ G }[/math], for [math]\displaystyle{ r\gt 0 }[/math], was given by Havelock 1955 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]
It should be noted that this 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]
Linton and McIver 2001. An equivalent representation is due to Kim 1965 for [math]\displaystyle{ r\gt 0 }[/math], although implicitly given in the work of Havelock 1955, 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.
The expression due to Peter and Meylan 2004 is
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} H_0^{(1)}(\alpha r) + \frac{1}{\pi^2} \int\limits_0^{\infty} \Big( \cos \eta z + \frac{\alpha}{\eta} \sin \eta z \Big) \frac{\eta^2}{\eta^2+\alpha^2} \Big( \cos \eta c + \frac{\alpha}{\eta} \sin \eta c \Big) K_0(\eta r) d\eta. }[/math]
