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. It is based on the Frequency Domain Problem. The exact form of the Green function depends on whether we assume the solution is proportional to [math]\displaystyle{ \exp(i\omega t) }[/math] or [math]\displaystyle{ \exp(-i\omega t) }[/math].

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}G,\,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 assume [math]\displaystyle{ \exp(i\omega t) }[/math] through out this.

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

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

with [math]\displaystyle{ k_0 }[/math] being purely imaginary with positive 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.,

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

and are given by

[math]\displaystyle{ N_n = \sqrt{\frac{\cos(k_nh)\sin(k_nh)+k_nh}{2k_n}} }[/math]

The Green function as written needs to only satisfy the condition

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

We can expand the delta function as

[math]\displaystyle{ \delta(z-c)=\sum_{n=0}^\infty f_n(z)f_n(c). }[/math]

Therefore we can derive the equation

[math]\displaystyle{ \sum_{n=0}^\infty (\partial_x^2 - k_n^2 )a_n(x)f_n(z)= \delta(x-a)\sum_{n=0}^\infty f_n(z)f_n(c). }[/math]

so that we must solve

[math]\displaystyle{ (\partial_x^2 - k_n^2 )a_n(x) = \delta(x-a)f_n(c). }[/math]

This has solution

[math]\displaystyle{ a_n(x) = -\frac{e^{-|x-a|k_n}f_n(c)}{2 k_n}. }[/math]

The Green function can therefore be written as

[math]\displaystyle{ G(x) = \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{2 k_n}f_n(c)f_n(z) }[/math]

[math]\displaystyle{ = \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{2 k_n N_n^2} \cos(k_n(z+h))\cos(k_n(c+h)) }[/math]

It can be written using the expression for [math]\displaystyle{ N_n }[/math] as

[math]\displaystyle{ G(\mathbf{x},\mathbf{\zeta}) = \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{\cos(k_nh)\sin(k_nh)+k_nh} \cos(k_n(z+h))\cos(k_n(c+h)) }[/math]

We can use the Dispersion Relation for a Free Surface which the roots [math]\displaystyle{ k_n }[/math] satisfy to show that [math]\displaystyle{ k_\infty = - k_n\tan k_n h }[/math] and [math]\displaystyle{ k_\infty^2+k_n^2 = k_n^2\sec^2k_n h }[/math] so that we can write the Green function in the following forms

[math]\displaystyle{ G(\mathbf{x},\mathbf{\zeta}) = \sum_{n=0}^\infty \frac{e^{-|x-a|k_n}}{k_n/k_\infty \sin(k_nh) - k_nh} \cos(k_n(z+h))\cos(k_n(c+h)) }[/math]

or

[math]\displaystyle{ G(\mathbf{x},\mathbf{\zeta}) = \sum_{n=0}^\infty \frac{(k_\infty^2+k_n^2)e^{-|x-a|k_n}}{k_\infty - (k_\infty^2+k_n^2)k_nh } \cos(k_n(z+h))\cos(k_n(c+h)) }[/math]

There are some numerical advantages to these other forms. Note that the expression give in Mei 1983 and Wehausen and Laitone 1960 is incorrect (by a factor of -1).

Incident at an angle

In some situations the potential may have a simple [math]\displaystyle{ e^{i k_y y} }[/math] dependence (so that it is pseudo two-dimensional). This is used to allows waves to be incident at an angle. 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 except we have to include [math]\displaystyle{ k_y }[/math]

[math]\displaystyle{ G(\mathbf{x},\mathbf{\zeta}) = \sum_{n=0}^\infty -\frac{k_n}{\sqrt{k_n^2+k_y^2}} \frac{e^{-|x-a|\sqrt{k_n^2+k_y^2}}}{\cos(k_nh)\sin(k_nh)+k_nh} \cos(k_n(z+h))\cos(k_n(c+h)) }[/math]

Infinite Depth

The Green function for infinite depth can be derived from the expression for finite depth by taking the limit as [math]\displaystyle{ h\to\infty }[/math] and converting the sum to an integral using the Riemann sum. Alternatively, the expression can be derived using Fourier Tranform

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{ \begin{matrix} G(\mathbf{x};\mathbf{\xi}) & = & \frac{i}{2} \, \frac{k_\infty^2-k^2}{(k_\infty^2-k^2)h-k_\infty}\, \cosh k(z+h)\, \cosh k(c+h) \, H_0^{(1)}(k r) \\ & + & \frac{1}{\pi} \sum_{m=1}^{\infty} \frac{k_m^2+k_\infty^2}{(k_m^2+k_\infty^2)h-k_\infty}\, \cos k_m(z+h)\, \cos k_m(c+h) \, K_0(k_m r), \end{matrix} }[/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 calculate Bessel functions for complex argument 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+k_\infty^2}{(k_m^2+k_\infty^2)h-k_\infty}\, \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+k_\infty^2}{h(k_m^2+k_\infty^2)-k_\infty}\, \cos k_m(z+h) \cos k_m(c+h) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r_+) I_\nu(k_m r_-) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}, }[/math]

where [math]\displaystyle{ r_+=\mathrm{max}\{r,s\} }[/math], and [math]\displaystyle{ r_-=\mathrm{min}\{r,s\} }[/math]; this was given by Black 1975 and Fenton 1978 and can be derived by applying Graf's Addition Theorem to [math]\displaystyle{ K_0(k_m|r\mathrm{e}^{\mathrm{i}\theta}-s\mathrm{e}^{\mathrm{i}\varphi}|) }[/math] in the definition of [math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) }[/math] above.

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{ \begin{matrix} G(\mathbf{x};\mathbf{\xi}) &= \frac{i k_\infty}{2} e^{k_\infty (z+c)} \, H_0^{(1)}(k_\infty r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} \\ & - \frac{1}{\pi^2} \int\limits_{0}^{\infty} \frac{k_\infty}{\eta^2 + k_\infty^2} \big( k_\infty \cos \eta (z+c) - \eta \sin \eta (z+c) \big) K_0(\eta r) d\eta. \end{matrix} }[/math]

It should be noted that this Green function can also be written in the following closely related form,

[math]\displaystyle{ \begin{matrix} G(\mathbf{x};\mathbf{\xi}) & = \frac{i k_\infty}{2} e^{k_\infty (z+c)} \, H_0^{(1)}(k_\infty r) + \frac{1}{4 \pi R_0} \\ & + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty} \frac{(\eta^2 - k_\infty^2) \cos \eta (z+c) + 2 \eta k_\infty \sin \eta (z+c)}{\eta^2 + k_\infty^2} K_0(\eta r) d\eta \end{matrix} }[/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{k_\infty}{4} e^{k_\infty (z+c)} \Big(\mathbf{H}_0(k_\infty r) + Y_0(k_\infty r) - 2i J_0 (k_\infty r) + \frac{2}{\pi} \int\limits_{z+c}^0 \frac{e^{-k_\infty \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 k_\infty}{2} e^{k_\infty (z+c)} h_0^{(1)}(k_\infty r) + \frac{1}{\pi^2} \int\limits_0^{\infty} \Big( \cos \eta z + \frac{k_\infty}{\eta} \sin \eta z \Big) \frac{\eta^2}{\eta^2+k_\infty^2} \Big( \cos \eta c + \frac{k_\infty}{\eta} \sin \eta c \Big) K_0(\eta r) d\eta. }[/math]