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 7: 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 36: 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 49: Line 47:
 
[[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 positive imaginary part and
+
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
 
<math>k_n,</math> <math>n\geq 1</math> are purely real with positive real part ordered with
 
increasing size.  
 
increasing size.  
 
<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
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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>
 
<center>
 
<center>
Line 95: Line 93:
 
<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))
Line 101: Line 99:
 
</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 = - k_n\tan k_n h</math>
+
<math>k_n</math> satisfy to show that <math>\alpha  = - k_n\tan k_n h</math>
 
and
 
and
<math>k_\infty^2+k_n^2 = k_n^2\sec^2k_n h</math>
+
<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
 
so that we can write the Green function in the following forms
 
<center>
 
<center>
 
<math>
 
<math>
G(\mathbf{x},\mathbf{\zeta})
+
G(\mathbf{x},\mathbf{\xi})
= \sum_{n=0}^\infty \frac{e^{-|x-a|k_n}}{k_n/k_\infty \sin(k_nh) - k_nh}
+
= \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))
 
\cos(k_n(z+h))\cos(k_n(c+h))
 
</math>
 
</math>
Line 115: Line 113:
 
<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>
Line 123: Line 121:
 
and [[Wehausen and Laitone 1960]] is incorrect (by a factor of -1).
 
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
Line 150: Line 148:
 
<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}
Line 157: Line 155:
 
</center>
 
</center>
  
== Infinite Depth ==
+
=== 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]]
+
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 ==
+
=== Solution for the singularity at the Free-Surface ===
  
 
We can also consider the following problem  
 
We can also consider the following problem  
 
<center>
 
<center>
 
<math>
 
<math>
\nabla^{2} G=0, \, -\infty<z<0
+
\nabla^{2} G=0, \, -h<z<0
 
</math>
 
</math>
 
</center>
 
</center>
Line 184: Line 182:
 
<center>
 
<center>
 
<math>
 
<math>
G(\mathbf{x},\mathbf{\zeta})
+
G(\mathbf{x},\mathbf{\xi})
 
= \sum_{n=0}^\infty -\frac{e^{-|x-a|k_n}}{2 k_n N_n^2}
 
= \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)
 
\cos(k_n(z+h))\cos(k_n h)
= \sum_{n=0}^\infty \frac{e^{-|x-a|k_n}}{2 k_\infty N_n^2}
+
= \sum_{n=0}^\infty \frac{e^{-|x-a|k_n}}{2 \alpha  N_n^2}
 
\cos(k_n(z+h))\sin(k_n h)
 
\cos(k_n(z+h))\sin(k_n h)
 
</math>
 
</math>
 
</center>
 
</center>
  
= Three Dimensional Representations =
+
=== 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 {{Green function source and field free surface code}}
 +
 
 +
== Three Dimensional Representations ==
  
 
Let <math>(r,\theta)</math> be cylindrical coordinates such that  
 
Let <math>(r,\theta)</math> be cylindrical coordinates such that  
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(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 223: Line 226:
 
\begin{matrix}
 
\begin{matrix}
 
G(\mathbf{x};\mathbf{\xi}) & = & \frac{i}{2} \,
 
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
+
\frac{\alpha ^2-k^2}{(\alpha ^2-k^2)h-\alpha }\, \cosh k(z+h)\, \cosh
 
k(c+h) \, H_0^{(1)}(k r) \\
 
k(c+h) \, H_0^{(1)}(k r) \\
 
  & + & \frac{1}{\pi} \sum_{m=1}^{\infty}
 
  & + & \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
+
\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),
 
k_m(c+h) \, K_0(k_m r),
 
\end{matrix}
 
\end{matrix}
 
</math>
 
</math>
 
</center>
 
</center>
where <math>H^{(1)}_0</math> and <math>K_0</math> denote the Hankel function of the first
+
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
 
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]]  
 
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]]  
Line 241: Line 244:
 
<math>
 
<math>
 
G(\mathbf{x};\mathbf{\xi}) = \frac{1}{\pi} \sum_{m=0}^{\infty}
 
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
+
\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),
 
k_m(c+h) \, K_0(k_m r),
 
</math>
 
</math>
Line 251: Line 254:
 
<center><math>
 
<center><math>
 
G(r,\theta,z;s,\varphi,c)=  \frac{1}{\pi} \sum_{m=0}^{\infty}
 
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
+
\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
 
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)},
 
(\theta - \varphi)},
Line 258: Line 261:
 
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.
 
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]] ==
+
=== [[Infinite Depth]] ===
  
 
In three dimensions and infinite depth the Green function <math>G</math>, for <math>r>0</math>, was
 
In three dimensions and infinite depth the Green function <math>G</math>, for <math>r>0</math>, was
Line 265: Line 268:
 
<math>
 
<math>
 
\begin{matrix}
 
\begin{matrix}
G(\mathbf{x};\mathbf{\xi}) &= \frac{i k_\infty}{2} e^{k_\infty (z+c)}
+
G(\mathbf{x};\mathbf{\xi}) &= \frac{i \alpha }{2} e^{\alpha  (z+c)}
\, H_0^{(1)}(k_\infty r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} \\
+
\, 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{1}{\pi^2} \int\limits_{0}^{\infty}
\frac{k_\infty}{\eta^2 + k_\infty^2} \big( k_\infty \cos \eta (z+c) - \eta \sin
+
\frac{\alpha }{\eta^2 + \alpha ^2} \big( \alpha  \cos \eta (z+c) - \eta \sin
\eta (z+c) \big)  K_0(\eta r) d\eta.
+
\eta (z+c) \big)  K_0(\eta r) \mathrm{d}\eta.
 
\end{matrix}
 
\end{matrix}
 
</math>
 
</math>
Line 278: Line 281:
 
<math>
 
<math>
 
\begin{matrix}
 
\begin{matrix}
G(\mathbf{x};\mathbf{\xi}) & = \frac{i k_\infty}{2}  e^{k_\infty (z+c)}
+
G(\mathbf{x};\mathbf{\xi}) & = \frac{i \alpha }{2}  e^{\alpha  (z+c)}
\, H_0^{(1)}(k_\infty r) + \frac{1}{4 \pi R_0} \\
+
\, H_0^{(1)}(\alpha  r) + \frac{1}{4 \pi R_0} \\
 
& + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty}
 
& + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty}
\frac{(\eta^2 - k_\infty^2) \cos \eta (z+c) + 2 \eta k_\infty \sin
+
\frac{(\eta^2 - \alpha ^2) \cos \eta (z+c) + 2 \eta \alpha  \sin
\eta (z+c)}{\eta^2 + k_\infty^2}  K_0(\eta r) d\eta  
+
\eta (z+c)}{\eta^2 + \alpha ^2}  K_0(\eta r) \mathrm{d}\eta  
 
\end{matrix}
 
\end{matrix}
 
</math>
 
</math>
Line 292: Line 295:
 
<math>
 
<math>
 
G(\mathbf{x};\mathbf{\xi}) = \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1}
 
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) +
+
- \frac{\alpha }{4} e^{\alpha  (z+c)} \Big(\mathbf{H}_0(\alpha  r) +
Y_0(k_\infty r) - 2i J_0 (k_\infty r)  + \frac{2}{\pi}
+
Y_0(\alpha  r) - 2i J_0 (\alpha  r)  + \frac{2}{\pi}
\int\limits_{z+c}^0 \frac{e^{-k_\infty \eta}}{\sqrt{r^2 + \eta^2}}
+
\int\limits_{z+c}^0 \frac{e^{-\alpha  \eta}}{\sqrt{r^2 + \eta^2}}
d\eta \Big),  
+
\mathrm{d}\eta \Big),  
 
</math>
 
</math>
 
</center>
 
</center>
Line 305: Line 308:
 
<center>
 
<center>
 
<math>
 
<math>
G(\mathbf{x};\mathbf{\xi}) = \frac{i k_\infty}{2} e^{k_\infty (z+c)}
+
G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha }{2} e^{\alpha  (z+c)}
h_0^{(1)}(k_\infty r) + \frac{1}{\pi^2} \int\limits_0^{\infty} \Big( \cos
+
h_0^{(1)}(\alpha  r) + \frac{1}{\pi^2} \int\limits_0^{\infty} \Big( \cos
\eta z + \frac{k_\infty}{\eta} \sin \eta z \Big)
+
\eta z + \frac{\alpha }{\eta} \sin \eta z \Big)
\frac{\eta^2}{\eta^2+k_\infty^2} \Big( \cos \eta c  +
+
\frac{\eta^2}{\eta^2+\alpha ^2} \Big( \cos \eta c  +
\frac{k_\infty}{\eta} \sin \eta c \Big)  K_0(\eta r) d\eta.
+
\frac{\alpha }{\eta} \sin \eta c \Big)  K_0(\eta r) \mathrm{d}\eta.
 
</math>
 
</math>
 
</center>
 
</center>
 +
 
[[Category:Linear Water-Wave Theory]]
 
[[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]