Difference between revisions of "Free-Surface Green Function"

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= Introduction =
 
 
 
The Free-Surface Green function is one of the most important objects in linear
 
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
 
water wave theory. It forms the basis on many of the numerical solutions, especially
Line 8: Line 5:
 
depends on whether we assume the solution is proportional to <math>\exp(i\omega t)</math>
 
depends on whether we assume the solution is proportional to <math>\exp(i\omega t)</math>
 
or <math>\exp(-i\omega t)</math>.
 
or <math>\exp(-i\omega t)</math>.
 
+
It is the fundamental tool for the [[Green Function Solution Method]]
 
There are many different representations for the Green function.
 
There are many different representations for the Green function.
  
= Equations 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]])
 
The Free-Surface Green function is a function which satisfies the following equation (in [[Finite Depth]])
 
<center>
 
<center>
 
<math>
 
<math>
\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\infty<z<0
+
\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -h<z<0
 
</math>
 
</math>
 
</center>
 
</center>
Line 26: Line 23:
 
<center>
 
<center>
 
<math>
 
<math>
  \frac{\partial G}{\partial z} = k_{\infty}G,\,z=0.
+
  \frac{\partial G}{\partial z} = \alpha G,\,z=0.
 
</math>
 
</math>
 
</center>
 
</center>
where <math>k_{\infty}</math> is the wavenumber in [[Infinite Depth]] which is given by  
+
where <math>\alpha</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
+
<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
 
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>
 
on whether we assume that the solution is proportional to <math>\exp(i\omega t)</math>
Line 37: Line 34:
 
We define <math>\mathbf{x}=(x,y,z)</math> and <math>\mathbf{\xi}=(a,b,c)</math>
 
We define <math>\mathbf{x}=(x,y,z)</math> and <math>\mathbf{\xi}=(a,b,c)</math>
  
= Two Dimensional Representations =  
+
== 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
 
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
Line 45: Line 42:
 
where
 
where
 
<center>
 
<center>
<math>f_n(z)=\frac{\cos(k_n(z h))}{N_n}</math>
+
<math>f_n(z)=\frac{\cos(k_n(z+h))}{N_n}</math>
 
</center>
 
</center>
 
<math>k_n</math> are the roots of the
 
<math>k_n</math> are the roots of the
 
[[Dispersion Relation for a Free Surface]]
 
[[Dispersion Relation for a Free Surface]]
 
<center><math>
 
<center><math>
k_\infty k_n\tan{(k_n h)}=  0\,
+
\alpha +k_n\tan{(k_n h)}=  0\,
 
</math></center>
 
</math></center>
 
with <math>k_0</math> being purely imaginary with negative imaginary part and
 
with <math>k_0</math> being purely imaginary with negative imaginary part and
Line 57: Line 54:
 
<math>N_n</math> is chosen so that the eigenfunctions are orthonormal, i.e.,
 
<math>N_n</math> is chosen so that the eigenfunctions are orthonormal, i.e.,
 
<center><math>
 
<center><math>
\int_{-h}^{0} f_m(z) f_n(z)dz = \delta_{mn}.\,
+
\int_{-h}^{0} f_m(z) f_n(z)\mathrm{d}z = \delta_{mn}.\,
 
</math></center>
 
</math></center>
 
and are given by
 
and are given by
 
<center><math>
 
<center><math>
N_n = \sqrt{\frac{\cos(k_nh)\sin(k_nh) k_nh}{2k_n}}
+
N_n = \sqrt{\frac{\cos(k_nh)\sin(k_nh)+k_nh}{2k_n}}
 
</math></center>
 
</math></center>
 
The Green function as written needs to only satisfy the condition
 
The Green function as written needs to only satisfy the condition
 
<center><math>
 
<center><math>
(\partial_x^2   \partial_z^2 )G = \delta(x-a)\delta(z-c).
+
(\partial_x^2 + \partial_z^2 )G = \delta(x-a)\delta(z-c).
 
</math></center>
 
</math></center>
 
We can expand the delta function as  
 
We can expand the delta function as  
Line 85: Line 82:
 
The Green function can therefore be written as
 
The Green function can therefore be written as
 
<center>
 
<center>
<math>G(x) = \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{2 k_n}f_n(c)f_n(z)</math>
+
<math>G(x,\xi) = \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>
 
</center>
 
It can be written using the expression for <math>N_n</math> as
 
It can be written using the expression for <math>N_n</math> as
 
<center>
 
<center>
 
<math>
 
<math>
G(\mathbf{x},\mathbf{\zeta})
+
G(\mathbf{x},\mathbf{\xi})
= \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{\cos(k_nh)\sin(k_nh) k_nh}
+
= \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))
+
\cos(k_n(z+h))\cos(k_n(c+h))
 
</math>
 
</math>
 
</center>
 
</center>
 
We can use the [[Dispersion Relation for a Free Surface]] which the roots
 
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>
+
<math>k_n</math> satisfy to show that <math>\alpha  = - k_n\tan k_n h</math>
so that we can write the Green function as
+
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{\xi})
 +
= \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>
 
<center>
 
<math>
 
<math>
G(\mathbf{x},\mathbf{\zeta})
+
G(\mathbf{x},\mathbf{\xi})
= \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 }
+
= \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))
+
\cos(k_n(z+h))\cos(k_n(c+h))
 
</math>
 
</math>
 
</center>
 
</center>
This form is numerically advantageous.
+
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 ==
+
=== Incident at an angle ===
  
 
In some situations the potential may have a simple <math>e^{i k_y y}</math> dependence
 
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
+
(so that it is pseudo two-dimensional). This is used to allow waves to be incident
 
at an angle.  
 
at an angle.  
 
We require the Green function to satisfy the following equation
 
We require the Green function to satisfy the following equation
 
<center>
 
<center>
 
<math>
 
<math>
\left(\partial_x^2   \partial_z^2 - k_y^2\right)
+
\left(\partial_x^2 + \partial_z^2 - k_y^2\right)
 
G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\infty<z<0
 
G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -\infty<z<0
 
</math>
 
</math>
Line 126: Line 140:
 
<center>
 
<center>
 
<math>
 
<math>
  \frac{\partial G}{\partial z} = k_{\infty}\phi,\,z=0.
+
  \frac{\partial G}{\partial z} = \alpha\phi,\,z=0.
 
</math>
 
</math>
 
</center>
 
</center>
  
The Green function be derived exactly as before except we have to include
+
The Green function can be derived exactly as before except we have to include
 
<math>k_y</math>
 
<math>k_y</math>
 
<center>
 
<center>
 
<math>
 
<math>
G(\mathbf{x},\mathbf{\zeta})
+
G(\mathbf{x},\mathbf{\xi})
= \sum_{n=0}^\infty -\frac{k_n}{\sqrt{k_n^2 k_y^2}}
+
= \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}
+
\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))
+
\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_transform Fourier Transform] [[Mei 1983]]
 +
 
 +
=== Solution for the singularity at the Free-Surface ===
 +
 
 +
We can also consider the following problem
 +
<center>
 +
<math>
 +
\nabla^{2} G=0, \, -h<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{\xi})
 +
= \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>
 
</math>
 
</center>
 
</center>
  
== Infinite Depth ==
+
=== Matlab code ===
  
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]
+
Code to calculate the Green function in two dimensions without incident angle for point source and field
 +
point on the free surface can be found here {{Green function source and field free surface code}}
  
= Three Dimensional Representations =
+
== Three Dimensional Representations ==
  
Let <math>(r,\theta)</math> be spherical coordinates such that  
+
Let <math>(r,\theta)</math> be cylindrical coordinates such that  
 
<center>
 
<center>
 
<math>
 
<math>
Line 162: Line 212:
 
and the distance from the ''mirror'' source point
 
and the distance from the ''mirror'' source point
 
<math>\bar{\mathbf{\xi}} = (a,b,-c)</math> respectively,
 
<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
+
<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>.
+
(z+c)^2</math>.
  
==[[Finite Depth]]==
+
===[[Finite Depth]]===
  
 
The most important representation of the finite depth free
 
The most important representation of the finite depth free
Line 175: Line 225:
 
<math>
 
<math>
 
\begin{matrix}
 
\begin{matrix}
G(\mathbf{x};\mathbf{\xi})
+
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) \mathrm{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) \mathrm{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}}
 +
\mathrm{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) \mathrm{d}\eta.
 +
</math>
 +
</center>
 +
 
 +
[[Category:Linear Water-Wave Theory]]
 +
[[Category:Complete Pages]]

Latest revision as of 08:57, 19 August 2010

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]. It is the fundamental tool for the Green Function Solution Method 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}), \, -h\lt z\lt 0 }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z}=0, \, z=-h, }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z} = \alpha G,\,z=0. }[/math]

where [math]\displaystyle{ \alpha }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ \alpha=\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{ \alpha +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)\mathrm{d}z = \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,\xi) = \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{\xi}) = \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{ \alpha = - k_n\tan k_n h }[/math] and [math]\displaystyle{ \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

[math]\displaystyle{ G(\mathbf{x},\mathbf{\xi}) = \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]

or

[math]\displaystyle{ G(\mathbf{x},\mathbf{\xi}) = \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]

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 allow 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} = \alpha\phi,\,z=0. }[/math]

The Green function can be derived exactly as before except we have to include [math]\displaystyle{ k_y }[/math]

[math]\displaystyle{ G(\mathbf{x},\mathbf{\xi}) = \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 Transform Mei 1983

Solution for the singularity at the Free-Surface

We can also consider the following problem

[math]\displaystyle{ \nabla^{2} G=0, \, -h\lt z\lt 0 }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z}=0, \, z=-h, }[/math]

[math]\displaystyle{ -\frac{\partial G}{\partial z} + \alpha G = \delta(x-a),\,z=0. }[/math]

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.

[math]\displaystyle{ G(\mathbf{x},\mathbf{\xi}) = \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]

Matlab code

Code to calculate the Green function in two dimensions without incident angle for point source and field point on the free surface can be found here two_d_finite_depth_Green_surface.m

Three Dimensional Representations

Let [math]\displaystyle{ (r,\theta) }[/math] be cylindrical 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{\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]

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+\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}{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]

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 \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) \mathrm{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 \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) \mathrm{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{\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}} \mathrm{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) \mathrm{d}\eta. }[/math]