View source for Eigenfunction Matching for a Submerged Circular Dock

{{complete pages}}

== Introduction ==

We present here the theory for a submerged circular dock. The details of the method
can be found in [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and
[[Eigenfunction Matching for a Circular Dock]]

==Governing Equations==

We begin with the [[Frequency Domain Problem]].
We will use a cylindrical coordinate system, <math>(r,\theta,z)</math>,
assumed to have its origin at the centre of the circular
plate which has radius <math>a</math>. The water is assumed to have
constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, \,\, -h<z<0,
</math>
</center>
<center>
<math>
\phi_{z}=0, \,\, z=-h,
</math>
</center>
<center><math>
\phi_{z}=\alpha\phi, \,\, z=0,
</math></center>
<center>
<math>
\phi_{z}=0, \,\, z=-d,\,r<a
</math>
</center>
We must also apply the [[Sommerfeld Radiation Condition]]
as <math>r\rightarrow\infty</math>. The subscript <math>z</math>
denotes the derivative in <math>z</math>-direction.

==Solution Method==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>r<a</math>
and <math>r>a</math>.

{{separation of variables in cylindrical coordinates in finite depth}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{separation of variables for a submerged dock}}

{{free surface submerged plate relations}}

{{separation of variables in cylindrical coordinates}}

Since the solution must be bounded
we know that under the plate the solution will be a linear combination of
<math>I_{n}(y)</math> while outside the plate the solution will be a
linear combination of <math>K_{n}(y)</math>. 
The case <math>\mu_1 =0</math> is a special case and the solution under
the dock is <math>(r/a)^{|n|}</math>.

Therefore the potential can
be expanded as
<center>
<math>
\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n}
(k_{m}r)e^{i n\theta}\phi_{m}(z), \;\;r>a
</math>
</center>
and
<center>
<math>
\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}b_{1n}(r/a)^{|n|} e^{i n\theta}\chi_{1}(z)+
\sum_{n=-\infty}^{\infty}\sum_{m=0,m\neq 1}^{\infty}b_{mn}
I_{n}(\kappa_{m}r)e^{i n\theta}\chi_{m}(z), \;\;r<a
</math>
</center>
where <math>a_{mn}</math> and <math>b_{mn}</math>
are the coefficients of the potential in the open water and
the plate covered region respectively.

{{incident plane wave in cylindrical coordinates}}

===An infinite dimensional system of equations===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>r=a</math> have to be equal
for each angle and we obtain
<center>
<math>
I_{n}(k_{0}a)\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}
a_{mn} K_{n}(k_{m}a)\phi_{m}\left(  z\right) 
= b_{1n} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\chi_{m}(z)
</math>
</center>
and
<center>
<math>
k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left(  z\right)  +\sum
_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left(  z\right) 
 =b_{1n} \frac{|n|}{a} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}
b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)
</math>
</center>
for each <math>n</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z)</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
=b_{1n}B_{0l} + \sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml} 
</math>
</center>
and
<center>
<math>
k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime
}(k_{l}a)A_{l} 
 = b_{1n}B_{0l}\frac{|n|}{a} + 
\sum_{m=0,m\neq 1}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)B_{ml} 
</math>
</center>

== Numerical Solution ==

To solve the system of equations we set the upper limit of <math>l</math> to
be <math>M</math>.

== Matlab Code ==

A program to calculate the coefficients for circular dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/circle_submerged_dock_matching_one_n.m circle_submerged_dock_matching_one_n.m]
Note that this problem solves only for a single n. 
=== Additional code ===

This program requires
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
to run


[[Category:Problems with Cylindrical Symmetry]]
[[Category:Eigenfunction Matching Method]]
[[Category:Pages with Matlab Code]]
[[Category:Complete Pages]]