View source for Eigenfunction Matching for a Circular Dock

=Introduction=

We show here a solution for a dock on [[Finite Depth]] water, which is circular. This is the three-dimensional
analog of the [[Eigenfunction Matching for a Semi-Infinite 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,\,r>a,
</math></center>
<center>
<math>
\phi_{z}=0, \,\, z=0,\,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=

== Separation of variables==

We now separate variables, noting that since the problem has
circular symmetry we can write the potential as
<center>
<math>
\phi(r,\theta,z)=\zeta(z)\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta}
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0
</math>
</center>
so that:
<center>
<math>
\zeta=\cos\mu(z+H)
</math>
</center>
where the separation constant <math>\mu^{2}</math> must
satisfy the [[Dispersion Relation for a Free Surface]]
<center>
<math>
k\tan\left(  kh\right)  =-\alpha,\quad r>a
</math>
</center>
and 
<center>
<math>
\kappa\tan(\kappa h)=0,\quad
r<a
</math>
</center>
Note that we have set <math>\mu=k</math> under the free
surface and <math>\mu=\kappa</math> under the dock. We denote the
positive imaginary solution of the [[Dispersion Relation for a Free Surface]] 
by <math>k_{0}</math> and
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
second equation will be denoted by  
<math>\kappa_{m} = m\pi/h</math>, <math>m\geq 0</math>. 

We define
<center>
<math>
\phi_{m}\left(  z\right)  =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water region and
<center>
<math>
\psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
m\geq 0
</math>
</center>
as the vertical eigenfunction of the potential in the dock
covered region. For later reference, we note that:
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
^{2}k_{m}h}\right)
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
</math>
</center>
where
<center><math>
B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
\kappa_{m}h}{\left(  \cos k_{n}h\right)  \left(  k_{n}
^{2}-\kappa_{m}^{2}\right)  }
</math></center>
and
<center>
<math>
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h
</math></center>

We now solve for the function <math>\rho_{n}(r)</math>.
Using Laplace's equation in polar coordinates we obtain
<center>
<math>
\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r} 
\frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left( 
\frac{n^{2}}{r^{2}}+\mu^{2}\right)  \rho_{n}=0
</math>
</center>
where <math>\mu</math> is <math>k_{m}</math> or
<math>\kappa_{m},</math> depending on whether <math>r</math> is
greater or less than <math>a</math>. We can convert this equation to the
standard form by substituting <math>y=\mu r</math> (provided that
<math>\mu\neq 0</math>to obtain
<center>
<math>
y^{2}\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}y^{2}}+y\frac{\mathrm{d}\rho_{n}
}{\rm{d}y}-(n^{2}+y^{2})\rho_{n}=0
</math>
</center>
The solution of this equation is a linear combination of the
modified Bessel functions of order <math>n</math>, <math>I_{n}(y)</math> and
<math>K_{n}(y)</math> ([[Abramowitz and Stegun 1964]]). 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>\kappa_0 =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_{0n}(r/a)^{|n|} e^{i n\theta}\psi_{0}(z)+
\sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}b_{mn}
I_{n}(\kappa_{m}r)e^{i n\theta}\psi_{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 potential==

The incident potential is a wave of amplitude <math>A</math>
in displacement travelling in the positive <math>x</math>-direction. 
The incident potential can therefore be written as
<center>
<math>
\phi^{\mathrm{I}}  =\frac{A}{i\sqrt{\alpha}}e^{k_{0}x}\phi_{0}\left(
z\right) 
=\sum\limits_{n=-\infty}^{\infty}e_{n}I_{n}(k_{0}r)\phi_{0}\left(z\right)
e^{i n \theta}
</math>
</center>
where <math>e_{n}=A/\left(i\sqrt{\alpha}\right)</math>
(we retain the dependence on <math>n</math> for situations
where the incident potential might take another form).





==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>
e_{n}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_{0n} \psi_{0}(z) +\sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)
</math>
</center>
and
<center>
<math>
 e_{n}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_{0n} \frac{|n|}{a} \psi_{0}(z) +\sum_{m=1}^{\infty}
b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\psi
_{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>
e_{n}I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
=b_{0n}B_{0l} + \sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml} 
</math>
</center>
and
<center>
<math>
e_{n}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_{0n}B_{0l}\frac{|n|}{a} + 
\sum_{m=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_dock_matching_one_n.m circle_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]]