Free-Surface Green Function

2007-04-11T10:58:53Z

AbxD5a:

= 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>\exp(i\omega t)</math>
or <math>\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]])
<center>
<math>
\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\infty<z<0
</math>
</center>
<center>
<math>
\frac{\partial G}{\partial z}=0, \, z=-h,
</math>
</center>
<center>
<math>
\frac{\partial G}{\partial z} = k_{\infty}\phi,\,z=0.
</math>
</center>
where <math>k_{\infty}</math> is the wavenumber in [[Infinite Depth]] which is given by
<math>k_{\infty}=\omega^2/g</math> where <math>g</math> is gravity. We also require a condition
as <math>\mathbf{x} \to \infty</math> which is the [[Sommerfeld Radiation Condition]]. This depends
on whether we assume that the solution is proportional to <math>\exp(i\omega t)</math>
or <math>\exp(-i\omega t)</math>. We assume <math>\exp(i\omega t)</math> through out this.

We define <math>\mathbf{x}=(x,y,z)</math> and <math>\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 [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]]. We write the Green function as
<center>
<math>G(x) = \sum_{n=0}^\infty a_n(x)f_n(z)</math>
</center>
where
<center>
<math>f_n(z)=\frac{\cos(k_n(z h))}{N_n}</math>
</center>
<math>k_n</math> are the roots of the
[[Dispersion Relation for a Free Surface]]
<center><math>
k_\infty k_n\tan{(k_n h)}= 0\,
</math></center>
with <math>k_0</math> being purely imaginary with negative imaginary part and
<math>k_n,</math> <math>n\geq 1</math> are purely real with positive real part ordered with
increasing size.
<math>N_n</math> is chosen so that the eigenfunctions are orthonormal, i.e.,
<center><math>
\int_{-h}^{0} f_m(z) f_n(z)dz = \delta_{mn}.\,
</math></center>
and are given by
<center><math>
N_n = \sqrt{\frac{\cos(k_nh)\sin(k_nh) k_nh}{2k_n}}
</math></center>
The Green function as written needs to only satisfy the condition
<center><math>
(\partial_x^2 \partial_z^2 )G = \delta(x-a)\delta(z-c).
</math></center>
We can expand the delta function as
<center><math>
\delta(z-c)=\sum_{n=0}^\infty f_n(z)f_n(c).
</math></center>
Therefore we can derive the equation
<center><math>
\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></center>
so that we must solve
<center><math>
(\partial_x^2 - k_n^2 )a_n(x) = \delta(x-a)f_n(c).
</math></center>
This has solution
<center><math>
a_n(x) = -\frac{e^{-|x-a|k_n}f_n(c)}{2 k_n}.
</math></center>
The Green function can therefore be written as
<center>
<math>G(x) = \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{2 k_n}f_n(c)f_n(z)</math>
</center>
It can be written using the expression for <math>N_n</math> as
<center>
<math>
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>
</center>
We can use the [[Dispersion Relation for a Free Surface]] which the roots
<math>k_n</math> satisfy to show that <math>k_\infty^2 k_n^2 = \sec^2k_n h</math>
so that we can write the Green function as
<center>
<math>
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>
</center>
This form is numerically advantageous.

==Incident at an angle ==

In some situations the potential may have a simple <math>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
<center>
<math>
\left(\partial_x^2 \partial_z^2 - k_y^2\right)
G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\infty<z<0
</math>
</center>
<center>
<math>
\frac{\partial G}{\partial z}=0, \, z=-h,
</math>
</center>
<center>
<math>
\frac{\partial G}{\partial z} = k_{\infty}\phi,\,z=0.
</math>
</center>

The Green function be derived exactly as before except we have to include
<math>k_y</math>
<center>
<math>
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>
</center>

== Infinite Depth ==

The Green function for infinite depth can be derived from the expression for finite depth by taking the limit as <math>h\to\infty</math> and converting the sum to an integral using the [http://en.wikipedia.org/wiki/Riemann_Sum Riemann sum]. Alternatively, the expression can be derived using [http://en.wikipedia.org/wiki/Fourier_tranform Fourier Tranform]

= Three Dimensional Representations =

Let <math>(r,\theta)</math> be spherical coordinates such that
<center>
<math>
x - a = r \cos \theta,\,
</math>
</center>
<center>
<math>
y - b = r \sin \theta,\,
</math>
</center>
and let <math>R_0</math> and <math>R_1</math> denote the
distance from the source point <math>\mathbf{\xi} = (a,b,c)</math>
and the distance from the ''mirror'' source point
<math>\bar{\mathbf{\xi}} = (a,b,-c)</math> respectively,
<math>R_0^2 = (x-a)^2 (y-b)^2 (z-c)^2</math> and <math>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_1950a|John 1950]]. He wrote the Green function in the
following form

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

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Free-Surface Green Function