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
 +
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>.
 +
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 =
+
== 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 16: Line 23:
 
<center>
 
<center>
 
<math>
 
<math>
  \frac{\partial G}{\partial z} = k_{\infty}\phi,\,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]]
+
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>
 
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 31: Line 40:
 
<math>G(x) = \sum_{n=0}^\infty a_n(x)f_n(z)</math>
 
<math>G(x) = \sum_{n=0}^\infty a_n(x)f_n(z)</math>
 
</center>
 
</center>
 +
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>
where <math>k_(n)</math> is the solution 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>
\alpha+k_n\tan{(k_n h)}=  0\,
+
\alpha +k_n\tan{(k_n h)}=  0\,
 
</math></center>
 
</math></center>
and <math>N_n</math> is chosen so that the eigenfunctions are orthonormal, i.e.,
+
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>
 
<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>
It can be written
+
and are given by
as
+
<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,\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>
 +
It can be written using the expression for <math>N_n</math> as
 +
<center>
 +
<math>
 +
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>
 +
</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>
 
<center>
<math>-\sum_{n=0}^\infty \frac{ie^{i k_n}}{\tan(k_m H) + H k_m\sec(k_m)^2}</math>
+
<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>
 
</center>
([[Linton_McIver_2001a| Linton and McIver 2001]])
+
or
 +
<center>
 +
<math>
 +
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>
 +
</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 ==
+
=== Incident at an angle ===
  
In some situations the potential may have a simple <math>e^{i k_y y}</math>
+
In some situations the potential may have a simple <math>e^{i k_y y}</math> dependence
dependence and we require the Green function to satisfy the following equation
+
(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>
 
<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 67: 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>
 +
</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{\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>
 +
</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>
 
</math>
 
</center>
 
</center>
  
= Three Dimensional Representations =
+
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>
 +
</center>
  
Let <math>(r,\theta)</math> be spherical coordinates such that
+
=== 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
 +
<center>
 
<math>
 
<math>
 
x - a = r \cos \theta,\,
 
x - a = r \cos \theta,\,
 
</math>
 
</math>
 
+
</center>
 +
<center>
 
<math>
 
<math>
 
y - b  = r \sin \theta,\,
 
y - b  = r \sin \theta,\,
 
</math>
 
</math>
 
+
</center>
 
and let <math>R_0</math> and <math>R_1</math> denote the  
 
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>
 
distance from the source point <math>\mathbf{\xi} = (a,b,c)</math>
Line 90: Line 215:
 
(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 97: Line 222:
 
following form
 
following form
  
 +
<center>
 
<math>
 
<math>
G(\mathbf{x};\mathbf{\xi}) = \frac{i}{2} \,
+
\begin{matrix}
\frac{\alpha^2-k^2}{(\alpha^2-k^2)H-\alpha}\, \cosh k(z+H)\, \cosh
+
G(\mathbf{x};\mathbf{\xi}) & = & \frac{i}{2} \,
k(c+H) \, H_0^{(1)}(k r)
+
\frac{\alpha ^2-k^2}{(\alpha ^2-k^2)h-\alpha }\, \cosh k(z+h)\, \cosh
  + \frac{1}{\pi} \sum_{m=1}^{\infty}
+
k(c+h) \, H_0^{(1)}(k r) \\
\frac{k_m^2+\alpha^2}{(k_m^2+\alpha^2)H-\alpha}\, \cos k_m(z+H)\, \cos
+
  & + & \frac{1}{\pi} \sum_{m=1}^{\infty}
k_m(c+H) \, K_0(k_m r),
+
\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>
 
</math>
 
+
</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]]  
 
and <math>k_m</math> are the imaginary parts of the solutions with positive imaginary part.  
 
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,
 
This way of writing the equation was primarily to avoid complex values for the Bessel functions,
however most computer packages will caculate Bessel functions for complex arguement so
+
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  
 
it makes more sense to write the Green function in the following form  
 
+
<center>
 
<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+\alpha^2}{(k_m^2+\alpha^2)H-\alpha}\, \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>
 
+
</center>
 
where <math>k_m</math> are as before except <math>k_0=ik</math>.
 
where <math>k_m</math> are as before except <math>k_0=ik</math>.
  
Line 126: 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+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos
+
\frac{k_m^2+\alpha ^2}{h(k_m^2+\alpha ^2)-\alpha }\, \cos k_m(z+h) \cos
k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \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)},
 
</math></center>
 
</math></center>
given by [[Black 1975]] and [[Fenton 1978]]
+
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]] ==
+
=== [[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
 
given by [[Havelock_1955a|Havelock 1955]] as
 
given by [[Havelock_1955a|Havelock 1955]] as
 
+
<center>
 
<math>
 
<math>
G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)}
+
\begin{matrix}
\, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1}
+
G(\mathbf{x};\mathbf{\xi}) &= \frac{i \alpha }{2} e^{\alpha (z+c)}
  - \frac{1}{\pi^2} \int\limits_{0}^{\infty}
+
\, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} \\
\frac{\alpha}{\eta^2 + \alpha^2} \big( \alpha \cos \eta (z+c) - \eta \sin
+
  & - \frac{1}{\pi^2} \int\limits_{0}^{\infty}
\eta (z+c) \big)  K_0(\eta r) d\eta.
+
\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>
 
</math>
 
+
</center>
 
It should be noted that this Green
 
It should be noted that this Green
 
function can also be written in the following closely related form,  
 
function can also be written in the following closely related form,  
 
+
<center>
 
<math>
 
<math>
G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2}  e^{\alpha (z+c)}
+
\begin{matrix}
\, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0}
+
G(\mathbf{x};\mathbf{\xi}) & = \frac{i \alpha }{2}  e^{\alpha (z+c)}
+ \frac{1}{2 \pi^2} \int\limits_{0}^{\infty}
+
\, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} \\
\frac{(\eta^2 - \alpha^2) \cos \eta (z+c) + 2 \eta \alpha \sin
+
& + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty}
\eta (z+c)}{\eta^2 + \alpha^2}  K_0(\eta r) d\eta  
+
\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>
 
</math>
 
+
</center>
 
[[Linton_McIver_2001a|Linton and McIver 2001]]. An equivalent representation is due to  
 
[[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  
 
[[Kim_1965a|Kim 1965]] for <math>r>0</math>, although implicitly given in the work of  
 
[[Havelock_1955a|Havelock 1955]], and is given by
 
[[Havelock_1955a|Havelock 1955]], and is given by
 
+
<center>
 
<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{\alpha}{4} e^{\alpha (z+c)} \Big(\mathbf{H}_0(\alpha r) +
+
- \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}
+
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}}
+
\int\limits_{z+c}^0 \frac{e^{-\alpha \eta}}{\sqrt{r^2 + \eta^2}}
d\eta \Big),  
+
\mathrm{d}\eta \Big),  
 
</math>
 
</math>
 
+
</center>
 
where <math>J_0</math> and <math>Y_0</math> are the Bessel functions  of order zero
 
where <math>J_0</math> and <math>Y_0</math> are the Bessel functions  of order zero
 
of the first and second
 
of the first and second
Line 173: Line 306:
  
 
The expression due to [[Peter_Meylan_2004b|Peter and Meylan 2004]] is
 
The expression due to [[Peter_Meylan_2004b|Peter and Meylan 2004]] is
 
+
<center>
 
<math>
 
<math>
G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)}
+
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
+
h_0^{(1)}(\alpha r) + \frac{1}{\pi^2} \int\limits_0^{\infty} \Big( \cos
\eta z + \frac{\alpha}{\eta} \sin \eta z \Big)
+
\eta z + \frac{\alpha }{\eta} \sin \eta z \Big)
\frac{\eta^2}{\eta^2+\alpha^2} \Big( \cos \eta c  +
+
\frac{\eta^2}{\eta^2+\alpha ^2} \Big( \cos \eta c  +
\frac{\alpha}{\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>
  
 
[[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]