View source for Green Function Method for a Finite Dock

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== Introduction ==

The problem of a two-dimensional finite dock is solved using a green function. 
The same problem is solved using eigenfunction matching in [[Eigenfunction Matching for a Finite Dock]].

== Equations for a Dock in the Frequency Domain ==

{{finite dock equations}}



== Equation in Terms of the Modes of the Plate ==

Under these assumptions, the equations become
<center><math>
\Delta\phi   =0,\,\,-h<z<0,
</math></center>
<center><math>
\partial_{z}\phi   =0,\,\,z=-h,
</math></center>
<center><math>
\alpha\phi  =\partial_{z}\phi,\,\,x\notin(-L,L),\ \ z=0,
</math></center>
<center><math>
i\omega\sum_{n=0}^{\infty}\zeta_{n}X_{n}    =\partial_{z}\phi,\,\,x\in
(-L,L),\,\, z=0,
</math></center>
<center><math>
\sum_{n=0}^{\infty}\zeta_{n}\left(  1+\beta\lambda_{n}^{4}\right)
X_{n}-\alpha\gamma\sum_{n=0}^{\infty}\zeta_{n}X_{n}    = -i\omega
\phi,\,\,x\in(-L,L),\,\, z=0.
</math></center>
We solve for the potential (and displacement) as the sum of
the diffracted and radiation potentials in the standard way,
as for a rigid body.
<center><math>
\phi=\phi^{\mathrm{D}}+\phi^{\mathrm{R}} ,\,
</math></center>
We begin with the diffraction potential <math>\phi^{\mathrm{D}}</math> which
satisfies the following equations
<center><math>
\Delta\phi^{\mathrm{D}}    =0,\,\,-h<z<0,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{D}}    =0,\,\,z=-h,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{D}}    =\alpha \phi^{\mathrm{D}},\,\,x\notin(-L,L),\,\,
z=0,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{D}}    =0,\,\,x\in(-L,L),\,\,z=0.
</math></center>
Furthermore, <math>\phi^{\mathrm{D}}</math>  satisfies the [[Sommerfeld Radiation Condition]]
<center><math>
\frac{\partial}{\partial x} \left(\phi^{\mathrm{D}}-\phi^{\rm
I} \right) \pm ik\left( \phi^{\mathrm{D}}-\phi^{\rm I}\right)
= 0
,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty,
</math></center>
where
<math>k ,\,</math> is the wavenumber,
which is the positive real solution of the [[Dispersion Relation for a Free Surface]]
<center><math>
k\tanh(kh)=\omega^{2} \,
</math></center>
and <math>\phi^{\rm I}</math> is the incident wave given by
<center><math>
\phi^{\rm I}  = \frac{\cosh(k(z+h))}{\cosh(kh)} e^{-i kx} \,
</math></center>
(which has unit amplitude in potential) and
is travelling towards positive
infinity.  


As the plate is floating on the surface, we can denote it as follows:
<center><math>
\phi^{\rm I}|_{z=0}  =  e^{-i kx} \,
</math></center>


We now consider the scattered potentials <math>\phi^{\mathrm{S}}</math>.  The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:
<center><math>
\phi^{\mathrm{D}}=\phi^{\mathrm{I}}+\phi^{\mathrm{S}} \,
</math></center>
from this, we can construct the following conditions:
<center><math>
\Delta\phi^{\mathrm{S}}    =0,\,\,-h<z<0,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{S}}    =0,\,\,z=-h,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{S}}    =\alpha\phi^{\mathrm{S}},\,\,x\notin(-L,L),\, \,
z=0
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{S}}    = -\partial_{z}\phi^{\mathrm{I}},\,\,x\in(-L,L),\,\,z=0.
</math></center> <br /><br />

We now consider the radiation potentials <math>\phi^{\mathrm{R}}</math>.  We can express the radiation potential as:
<center><math>
\phi^{\mathrm{R}}=\sum_{n=0}^{\infty}\zeta_n \phi_n^{\mathrm{R}}
</math></center>
which satisfy the following equations
<center><math>
\Delta\phi_n^{\mathrm{R}}    =0,\,\,-h<z<0,
</math></center>
<center><math>
\partial_{z}\phi_n^{\mathrm{R}}    =0,\,\,z=-h,
</math></center>
<center><math>
\partial_{z}\phi_n^{\mathrm{R}}    =\alpha\phi_n^{\mathrm{R}},\,\,x\notin(-L,L),\, \,
z=0
</math></center>
<center><math>
\partial_{z}\phi_n^{\mathrm{R}}    = i\omega X_{n},\,\,x\in(-L,L),\,\,z=0.
</math></center>
The radiation condition for the radiation potential is
<center><math>
\frac{\partial\phi_n^{\mathrm{R}}}{\partial x}\pm ik\phi_n^{\mathrm{R}}=0,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty.
</math></center>
Therefore we find the potential as
<center><math>
\phi=\phi^{\mathrm{D}} +\sum_{n=0}^{\infty}\zeta_{n}\phi_n^{\mathrm{R}},
</math></center>
so that
<center><math>
\sum_{n=0}^{\infty}\left( 1+\beta\lambda_{n}^{4} - \alpha\gamma\right)
\zeta_{n}X_{n}=-i\omega \left( \phi^{\mathrm{D}}+\sum_{n=0}^{\infty}\zeta_{n}\phi_n^{\mathrm{R}} \right).
</math></center>
If we multiply by <math>X_m</math> and take an inner product over the plate we obtain
<center><math>
\left(  1+\beta\lambda_{n}^{4} - \alpha\gamma\right)  \zeta_{n}=-i\omega
\int_{-L}^{L}\phi^{\mathrm{D}} X_{n}\mathrm{d}x +
\sum_{m=0}^{\infty}\left(\omega^2 a_{mn}(\omega) - i\omega b_{mn}(\omega)\right)
\zeta_{m},
</math></center>
where the functions <math>a_{mn}(\omega)</math> and <math>b_{mn}(\omega)</math> are given by
<center><math>
\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,
</math></center>
and they are referred to as the added mass and damping coefficients (see [[Added-Mass, Damping Coefficients And Exciting Forces]]
for the equivalent definition for a rigid body).
respectively.
This equation is solved by truncating the number of modes.

== Solution for the Radiation and Diffracted Potential ==

We use the [[Free-Surface Green Function]] for two-dimensional waves, with singularity at
the water surface since we are only
interested in its value at <math>z=0.</math> Using this we can transform the system of equations to 

<center><math>
\phi^{\mathrm{D}}(x) = \phi^{\mathrm{I}}(x) + \int_{-L}^{L}G(x,\xi) \alpha\phi^{\mathrm{S}}(\xi) \mathrm{d} \xi
</math></center>
and 
<center><math>
\phi_n^{\mathrm{R}}(x) =  \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi_n^{\mathrm{R}}(\xi) - i\omega X_n(\xi)
\right)\mathrm{d} \xi
</math></center>

Details about this method can be found in [[Integral Equation for the Finite Depth Green Function at Surface]].

== Reflection and Transmission Coefficients ==

The Reflection and Transmission Coefficients represent the ratio of the amplitude of the reflected or transmitted wave to the amplitude of the incident wave.  They hold the property that <math> |R|^2+|T|^2=1\,</math> (and may often contain an imaginery element).

[[Image:Square_volume.png|600px|right|thumb|frame|A diagram depicting the area <math>\Omega\,</math> which is bounded by the rectangle <math>\partial \Omega \,</math>.  The rectangle <math>\partial \Omega \,</math> is bounded by <math> -h \leq z \leq 0 \,</math> and <math>-\infty  \leq x \leq  \infty \,</math> or <math>-N  \leq x \leq  N\,</math>]]

We can calculate the Reflection and Transmission coefficients as follows:
Applying Green's theorem to <math>\phi\,</math> and <math>\phi^{\mathrm{I}}\,</math> gives:
<center><math>
0 = \iint_{\Omega}(\phi\nabla^2\phi^{\mathrm{I}} - \phi^{\mathrm{I}}\nabla^2\phi)\mathrm{d}x\mathrm{d}z 
	= \int_{\partial\Omega}(\phi\frac{\partial\phi^{\rm I}}{\partial n} - \phi^{\rm I}\frac{\partial\phi}{\partial n})\mathrm{d}l,
</math></center>
<center><math>
	= \phi_0(0) \int_{-L}^{L} e^{k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x 
  - 2k_0 R  \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z.
</math></center><br\>
where  <math> k_0 \,</math> is the first imaginery root of the dispersion equation and the incident wave is of the form: <math> \phi^I=\phi_0(z)e^{-ikx} \,</math><br\><br\>
Therefore, in the case of a floating plate (where z=0):
<center><math>
R 	= \frac{\int_{-L}^{L} e^{k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x }
 {2 k_0  \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z}.
</math></center>
and using a wave incident from the right we obtain
<center><math>
1 + T 	= \frac{\int_{-L}^{L} e^{-k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x }
 {2 k_0 \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z}.
</math></center>

== Matlab Code ==


A program to calculate the solution in elastic modes can be found here
{{elastic plate modes code}}

=== Additional code ===

This program requires
* {{green function surface code}}
* {{green function source and field free surface code}}
* {{free surface dispersion equation code}}
* {{eigenvalues beam}}
* {{eigenvectors beam}}

== Alternative Solution Method using Green Functions for a Uniform Plate ==

We can also solve the equation by a closely related method which was given in 
[[Meylan and Squire 1994]].
We can transform the equations to
<center><math>
\phi(x) = \phi^{\rm I}(x) + \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi(\xi) - \partial_z\phi(\xi)
\right)\mathrm{d} \xi
</math></center>

Expanding as before
<center>
<math>
\partial_z \phi = i\omega \sum \xi_n X_n
</math>
</center>
we obtain
<center><math>
-i\omega \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)\xi_n X_n
</math>
</center>
This leads to the following equation 
<center>
<math>
\partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1} \phi(\xi)\mathrm{d}\xi
</math>
</center>
or 
<center>
<math>
\partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} g(x,\xi) \phi(\xi)\mathrm{d}\xi
</math>
</center>
where 
<center>
<math>
g(x,\xi) = \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}
</math>
</center>
which is the Green function for the plate.

[[Category:Floating Elastic Plate]]