View source for 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>\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} = \alpha G,\,z=0.
</math>
</center>
where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by 
<math>\alpha=\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>
\alpha +k_n\tan{(k_n h)}=  0\,
</math></center>
with <math>k_0</math> being purely imaginary with positive 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>
<center>
<math>
= \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>
</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>\alpha  = - k_n\tan k_n h</math>
and
<math>\alpha ^2+k_n^2 = k_n^2\sec^2k_n h</math>
so that we can write the Green function in the following forms
<center>
<math>
G(\mathbf{x},\mathbf{\zeta})
= \sum_{n=0}^\infty \frac{e^{-|x-a|k_n}}{k_n/\alpha  \sin(k_nh) - k_nh}
\cos(k_n(z+h))\cos(k_n(c+h))
</math>
</center>
or
<center>
<math>
G(\mathbf{x},\mathbf{\zeta})
= \sum_{n=0}^\infty \frac{(\alpha ^2+k_n^2)e^{-|x-a|k_n}}{\alpha  - (\alpha ^2+k_n^2)k_nh }
\cos(k_n(z+h))\cos(k_n(c+h))
</math>
</center>
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>e^{i k_y y}</math> dependence
(so that it is pseudo two-dimensional). This is used to allow 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} = \alpha\phi,\,z=0.
</math>
</center>

The Green function can 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] [[Mei 1983]]

== Solution for the singularity at the Free-Surface ==

We can also consider the following problem 
<center>
<math>
\nabla^{2} G=0, \, -\infty<z<0
</math>
</center>
<center>
<math>
\frac{\partial G}{\partial z}=0, \, z=-h,
</math>
</center>
<center>
<math>
 -\frac{\partial G}{\partial z} + \alpha G = \delta(x-a),\,z=0.
</math>
</center>

It turns out that the solution to this is nothing more than the Green function we found previously restricted
to the free surface, i.e.
<center>
<math>
G(\mathbf{x},\mathbf{\zeta})
= \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{2 k_n N_n^2}
\cos(k_n(z+h))\cos(k_n h)
= \sum_{n=0}^\infty \frac{e^{-|x-a|k_n}}{2 \alpha  N_n^2}
\cos(k_n(z+h))\sin(k_n h)
</math>
</center>

= Three Dimensional Representations =

Let <math>(r,\theta)</math> be cylindrical 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}) & = & \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),
\end{matrix}
</math>
</center>
where <math>H^{(1)}_0</math> and <math>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_Stegun_1964a|Abramowitz and Stegun 1964]], <math>k</math> is the positive real solution to the [[Dispersion Relation for a Free Surface]] 
and <math>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 
<center>
<math>
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>
</center>
where <math>k_m</math> are as before except <math>k_0=ik</math>.

An expression where both variables are given in cylindrical polar coordinates
is the following
<center><math>
G(r,\theta,z;s,\varphi,c)=  \frac{1}{\pi} \sum_{m=0}^{\infty}
\frac{k_m^2+\alpha ^2}{h(k_m^2+\alpha ^2)-\alpha }\, \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></center>
where <math>r_+=\mathrm{max}\{r,s\}</math>, and <math>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>K_0(k_m|r\mathrm{e}^{\mathrm{i}\theta}-s\mathrm{e}^{\mathrm{i}\varphi}|)</math> in the definition of <math>G(\mathbf{x};\mathbf{\xi})</math> above.

== [[Infinite Depth]] ==

In three dimensions and infinite depth the Green function <math>G</math>, for <math>r>0</math>, was
given by [[Havelock_1955a|Havelock 1955]] as
<center>
<math>
\begin{matrix}
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.
\end{matrix}
</math>
</center>
It should be noted that this Green
function can also be written in the following closely related form, 
<center>
<math>
\begin{matrix}
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 
\end{matrix}
</math>
</center>
[[Linton_McIver_2001a|Linton and McIver 2001]]. An equivalent representation is due to 
[[Kim_1965a|Kim 1965]] for <math>r>0</math>, although implicitly given in the work of 
[[Havelock_1955a|Havelock 1955]], and is given by
<center>
<math>
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>
</center>
where <math>J_0</math> and <math>Y_0</math> are the Bessel functions  of order zero
of the first and second
kind and <math>\mathbf{H}_0</math> is the Struve function of order zero.

The expression due to [[Peter_Meylan_2004b|Peter and Meylan 2004]] is
<center>
<math>
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>
</center>
[[Category:Linear Water-Wave Theory]]