Eigenfunction Matching for a Submerged Circular Dock: Difference between revisions
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as <math>r\rightarrow\infty</math>. The subscript <math>z</math> | as <math>r\rightarrow\infty</math>. The subscript <math>z</math> | ||
denotes the derivative in <math>z</math>-direction. | denotes the derivative in <math>z</math>-direction. | ||
= Separation of Variables = | |||
We can separate variables and we obtain the following expression for the velocity potential | |||
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. | |||
Revision as of 03:25, 25 July 2008
Introduction
We present here very briefly 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 Submerged Finite Dock
Governing Equations
We begin with the Frequency Domain Problem. We will use a cylindrical coordinate system, [math]\displaystyle{ (r,\theta,z) }[/math], assumed to have its origin at the centre of the circular plate which has radius [math]\displaystyle{ a }[/math]. The water is assumed to have constant finite depth [math]\displaystyle{ h }[/math] and the [math]\displaystyle{ z }[/math]-direction points vertically upward with the water surface at [math]\displaystyle{ z=0 }[/math] and the sea floor at [math]\displaystyle{ z=-h }[/math]. The boundary value problem can therefore be expressed as
[math]\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, }[/math]
[math]\displaystyle{ \phi_{z}=0, \,\, z=-h, }[/math]
[math]\displaystyle{ \phi_{z}=0, \,\, z=-d,\,r\lt a }[/math]
We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ r\rightarrow\infty }[/math]. The subscript [math]\displaystyle{ z }[/math] denotes the derivative in [math]\displaystyle{ z }[/math]-direction.
Separation of Variables
We can separate variables and we obtain the following expression for the velocity potential
Therefore the potential can be expanded as
[math]\displaystyle{ \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\gt a }[/math]
and
[math]\displaystyle{ \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\lt a }[/math]
where [math]\displaystyle{ a_{mn} }[/math] and [math]\displaystyle{ b_{mn} }[/math] are the coefficients of the potential in the open water and the plate covered region respectively.
