Free-Surface 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

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]

[math]\displaystyle{ f_n(z)=\frac{\cos(k_n(z+h)}{N_n} }[/math]

where [math]\displaystyle{ k_(n) }[/math] is the solution of the Dispersion Relation for a Free Surface

[math]\displaystyle{ \alpha+k_n\tan{(k_n h)}= 0\, }[/math]

and [math]\displaystyle{ N_n }[/math] is chosen so that the eigenfunctions are orthonormal, i.e.,

[math]\displaystyle{ \int_{-h}^{0} f_m(z) f_n(z)dz = \delta_{mn}.\, }[/math]

The Green function as written needs to only satisfy the condition

[math]\displaystyle{ (\partial_x^2 + \partial_z^2 )G = \delta(x-\xi)\delta(z-\eta) }[/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]

( Linton and McIver 2001)

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]

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

[math]\displaystyle{ G(r,\theta,z;s,\varphi,c)= \frac{1}{\pi} \sum_{m=0}^{\infty} \frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}, }[/math]

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]