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 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.,

[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]

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^2 k_n^2 = \sec^2k_n h }[/math] so that we can write the Green function as

[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]

This form is numerically advantageous.

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> \begin{matrix} G(\mathbf{x};\mathbf{\xi})