View source for Eigenfunction Matching for a Circular Floating Elastic Plate

{{complete pages}}

==Introduction==

We show here a solution for a [[Floating Elastic Plate]] on [[Finite Depth]] water
based on [[Peter_Meylan_Chung_2004a|Peter, Meylan and Chung 2004]]. A solution
for [[:Category:Shallow Depth|Shallow Depth]] was given in [[Zilman_Miloh_2000a|Zilman and Miloh 2000]] and we will also show this.
The solution is an extension of the [[Eigenfunction Matching for a Circular Dock]].

==Governing Equations==

We begin with the [[Frequency Domain Problem]] for a [[Floating Elastic Plate]]
in the non-dimensional form of [[Tayler_1986a|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]])
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>
(\beta\Delta^{2}+1-\alpha\gamma)\phi_{z}=\alpha\phi, \,\, z=0,\,r<a
</math>
</center>
where the constants <math>\beta</math> and <math>\gamma</math> are given by
<center>
<math>
\beta=\frac{D}{\rho\,L^{4}g}, \gamma=\frac{\rho_{i}h}{\rho\,L}
</math>
</center>
and <math>\rho_{i}</math> is the density of the plate. We
must also apply the edge conditions for the plate and 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 floating elastic plate}}

{{free surface floating elastic plate relations}}

{{separation of variables in cylindrical coordinates}}

{{incident plane wave in cylindrical coordinates}}

===Expansion of the potential ===

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>. 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}\sum_{m=-2}^{\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.
==Boundary conditions==

The boundary conditions for the plate also have to be
considered. The vertical force and bending moment must vanish, which can be
written as 
<center>
<math>
\left[\bar{\Delta}-\frac{1-\nu}{r}\left(\frac{\partial}{\partial r}
+\frac{1}{r}\frac{\partial^{2}}{\partial\theta^{2}}\right)\right]
w=0\,\,\,(3)
</math>
</center>
and
<center>
<math>
\left[  \frac{\partial}{\partial r}\bar{\Delta}-\frac{1-\nu}{r^{2}}\left(
-\frac{\partial}{\partial r}+\frac{1}{r}\right)  \frac{\partial^{2}}
{\partial\theta^{2}}\right]  w=0 \,\,\,(4)
</math>
</center>
where <math>w</math> is the time-independent surface
displacement, <math>\nu</math> is Poisson's ratio, and <math>\bar{\Delta}</math> is the
polar coordinate Laplacian
<center>
<math>
\bar{\Delta}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial
}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}
</math>
</center>

=== Displacement of the plate ===

The surface displacement and the water velocity potential at
the water surface are linked through the kinematic boundary condition
<center>
<math>
\phi_{z}=-i\sqrt{\alpha}w,\,\,\,z=0
</math>
</center>
From the equations the potential and the surface
displacement are therefore related by
<center>
<math>
w=i\sqrt{\alpha}\phi,\quad r>a
</math>
</center>
and
<center>
<math>
(\beta\bar{\Delta}^{2}+1-\alpha\gamma)w=i\sqrt{\alpha}\phi,\quad r<a
</math>
</center>
The surface displacement can also be expanded in eigenfunctions
as
<center>
<math>
w(r,\theta)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}i\sqrt{\alpha}
a_{mn}K_{n}(k_{m}r)e^{i n\theta},\;\;r>a
</math>
</center>
and:
<center>
<math>
w(r,\theta)=
\sum_{n=-\infty}^{\infty}\sum_{m=-2}^{\infty}i\sqrt{\alpha}(\beta\kappa
_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}I_{n}(\kappa_{m}r)e^{i
n\theta},\; r<a
</math>
</center>
using the fact that
<center>
<math>
\bar{\Delta}\left(  I_{n}(\kappa_{m}r)e^{i n\theta}\right)  =\kappa_{m}
^{2}I_{n}(\kappa_{m}r)e^{i n\theta}\,\,\,(5)
</math>
</center>

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

The boundary conditions (3) and
(4) can be expressed in terms of the potential
using (5). Since the angular modes are uncoupled the
conditions apply to each mode, giving
<center>
<math>
\sum_{m=-2}^{\infty}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
\left(\kappa_{m}^{2}I_{n}(\kappa_{m}a)-\frac{1-\nu}{a}\left(\kappa
_{m}I_{n}^{\prime}(\kappa_{m}a)-\frac{n^{2}}{a}I_{n}(\kappa_{m}a)\right)
\right) =0\,\,\,(6)
</math>
</center>
<center>
<math>
\sum_{m=-2}^{\infty}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
\left(  \kappa_{m}^{3}I_{n}^{\prime}(\kappa_{m}a)+n^{2}\frac{1-\nu}{a^{2}
}\left(  -\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)+\frac{1}{a}I_{n}(\kappa
_{m}a)\right)  \right)
=0\,\,\,(7)
</math>
</center>
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.
Again we know that this must be true 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) 
=\sum_{m=-2}^{\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) 
 =\sum_{m=-2}^{\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}
=\sum_{m=-2}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml} \,\,\,(8)
</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} 
 =\sum_{m=-2}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}
a)B_{ml} \,\,\,(9)
</math>
</center>
Equation (8) can be solved for the open water
coefficients <math>a_{mn}</math>
<center>
<math>
a_{ln}=-e_{n}\frac{I_{n}(k_{0}a)}{K_{n}(k_{0}a)}\delta_{0l}+\sum
_{m=-2}^{\infty}b_{mn}\frac{I_{n}(\kappa_{m}a)B_{ml}}{K_{n}(k_{l}a)A_{l}}
</math>
</center>
which can then be substituted into equation
(9) to give us
<center>
<math>
\left(  k_{0}I_{n}^{\prime}(k_{0}a)-k_{0}\frac{K_{n}^{\prime}(k_{0}
a)}{K_{n}(k_{0}a)}I_{n}(k_{0}a)\right)  e_{n}A_{0}\delta_{0l}
 =\sum_{m=-2}^{\infty}\left(  \kappa_{m}I_{n}^{\prime}(\kappa_{m}
a)-k_{l}\frac{K_{n}^{\prime}(k_{l}a)}{K_{n}(k_{l}a)}I_{n}(\kappa
_{m}a)\right)  B_{ml}b_{mn}\,\,\,(10)
</math>
</center>
for each <math>n</math>.
Together with equations (6) and (7)
equation (10) gives the required equations to solve for the
coefficients of the water velocity potential in the plate covered region.

==Numerical Solution==

To solve the system of equations (10) together
with the boundary conditions (6 and 7) we set the upper limit of <math>l</math> to
be <math>M</math>. We also set the angular expansion to be from 
<math>n=-N</math> to <math>N</math>. This gives us
<center>
<math>
\phi(r,\theta,z)=\sum_{n=-N}^{N}\sum_{m=0}^{M}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=-N}^{N}\sum_{m=-2}^{M}b_{mn}I_{n}(\kappa
_{m}r)e^{i n\theta}\psi_{m}(z), \;\;r<a 
</math>
</center>
Since <math>l</math> is an integer with <math>0\leq l\leq
M</math> this leads to a system of <math>M+1</math> equations.
The number of unknowns is <math>M+3</math> and the two extra equations
are obtained from the boundary conditions for the free plate (6)
and (7). The equations to be solved for each <math>n</math> are
<center>
<math>
\left(  k_{0}I_{n}^{\prime}(k_{0}a)-k_{0}\frac{K_{n}^{\prime}(k_{0}
a)}{K_{n}(k_{0}a)}I_{n}(k_{0}a)\right)  e_{n}A_{0}\delta_{0l}
=\sum_{m=-2}^{M}\left(  \kappa_{m}I_{n}^{\prime}(\kappa_{m}a)-k_{l}
\frac{K_{n}^{\prime}(k_{l}a)}{K_{n}(k_{l}a)}I_{n}(\kappa_{m}a)\right)
B_{ml}b_{mn} 
</math>
</center>
<center>
<math>
\sum_{m=-2}^{M}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
\left(  \kappa_{m}^{2}I_{n}(\kappa_{m}a)-\frac{1-\nu}{a}\left(  \kappa
_{m}I_{n}^{\prime}(\kappa_{m}a)-\frac{n^{2}}{a}I_{n}(\kappa_{m}a)\right)
\right) =0
</math>
</center>
and
<center>
<math>
\sum_{m=-2}^{M}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
\left(  \kappa_{m}^{3}I_{n}^{\prime}(\kappa_{m}a)+n^{2}\frac{1-\nu}{a^{2}
}\left(  \kappa_{m}I_{n}^{\prime}(\kappa_{m}a)+\frac{1}{a}I_{n}(\kappa
_{m}a)\right)  \right) =0 
</math>
</center>
It should be noted that the solutions for positive and negative
<math>n</math> are identical so that they do not both need to be
calculated. There are some minor simplifications which are a consequence of
this which are discussed in more detail in [[Zilman_Miloh 2000a|Zilman and Miloh 2000]].

==The [[:Category:Shallow Depth|Shallow Depth]] Theory of [[Zilman_Miloh_2000a|Zilman and Miloh 2000]]==

The shallow water theory of [[Zilman_Miloh_2000a|Zilman and Miloh 2000]] can be recovered by
simply setting the depth shallow enough that the shallow water theory is valid
and setting <math>M=0</math>. If the shallow water theory is valid then
the first three roots of the dispersion equation for the ice will be exactly
the same roots found in the shallow water theory by solving the polynomial
equation. The system of equations has four unknowns (three under the plate and
one in the open water) exactly as for the theory of [[Zilman_Miloh_2000a|Zilman and Miloh 2000]].

== A Simple Method To Calculate The [[Diffraction Transfer Matrix]] For The Case Of A Circular Plate ==

Let's consider an incident wave whose potential has the following expression
<center><math> 
\phi^\mathrm{I} (r,\theta,z) = \sum_{n=0}^{\infty} \phi_n(z)
\sum_{\nu = - \infty}^{\infty} D_{n\nu} I_\nu (k_n r) \mathrm{e}^{\mathrm{i}\nu \theta}.
</math></center>
Such an incident potential is found in the [[Kagemoto and Yue Interaction Theory]], where
it can be written as the sum of an ambient incident potential and the scattered potentials 
of the other bodies, which are interpretated as incident potentials for the studied body.

We can apply the same eigenfunction matching that previously, considering the potential 
and its normal derivative continuous at <math>r=a</math>. Thus the potential and its normal 
derivative expressed at each side of this value of the radius have to be equal. We obtain 
the relationships
<center><math>
\sum_{m=0}^{\infty} D_{mn} I_n (k_m a) \phi_m(z) + \sum_{m=0}^{\infty}
a_{mn} K_{n}(k_{m}a)\phi_{m}\left(  z\right) 
=\sum_{m=-2}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)
</math></center>
and 
<center><math>
\sum_{m=0}^{\infty} D_{mn} k_m I'_n (k_m a) \phi_m(z) + \sum_{m=0}^{\infty}
a_{mn} k_m K'_{n}(k_{m}a)\phi_{m}\left(  z\right) 
=\sum_{m=-2}^{\infty}b_{mn} \kappa_m I'_{n}(\kappa_{m}a)\psi_{m}(z)
</math></center>
for each <math>n</math>.
We solve these equations with the same method that before, by multiplying both equations by
<math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:
<center><math>
D_{ln} I_n (k_l a) A_l + a_{ln} K_{n}(k_l a) A_l 
= \sum_{m=-2}^{\infty} b_{mn} I_{n}(\kappa_{m} a)  B_{ml},\ \ \ (11)
</math></center>
and
<center><math>
D_{ln} k_l I'_n (k_l a) A_l + a_{ln} k_l K'_{n}(k_{l} a) A_l 
=\sum_{m=-2}^{\infty}b_{mn} \kappa_m I'_{n}(\kappa_{m}a) B_{ml} \ \ \ (12)
</math></center>

The [[Diffraction Transfer Matrix]] 
maps the coefficients of the incident wave with the coefficients of the scattered wave within 
the open water domain. The relation which links these two coefficients can be written as follows
<center><math>
a_{mn}=\sum_{l=0}^{\infty} T_{lmn} D_{ln}
</math></center>

Furthermore the boundary conditions are exactly the same that before, namely
<center>
<math>
\sum_{m=-2}^{\infty}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
\left(\kappa_{m}^{2}I_{n}(\kappa_{m}a)-\frac{1-\nu}{a}\left(\kappa
_{m}I_{n}^{\prime}(\kappa_{m}a)-\frac{n^{2}}{a}I_{n}(\kappa_{m}a)\right)
\right) =0
</math>
</center>
<center>
<math>
\sum_{m=-2}^{\infty}(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}b_{mn}
\left(  \kappa_{m}^{3}I_{n}^{\prime}(\kappa_{m}a)+n^{2}\frac{1-\nu}{a^{2}
}\left(  -\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)+\frac{1}{a}I_{n}(\kappa
_{m}a)\right)  \right)
=0 .
</math>
</center>
For the further study, let's call 
<center>
<math>
L^1_{mn}=(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}
\left(\kappa_{m}^{2}I_{n}(\kappa_{m}a)-\frac{1-\nu}{a}\left(\kappa
_{m}I_{n}^{\prime}(\kappa_{m}a)-\frac{n^{2}}{a}I_{n}(\kappa_{m}a)\right)
\right)
</math>
</center>
and
<center>
<math>
L^2_{mn}=(\beta\kappa_{m}^{4}+1-\alpha\gamma)^{-1}
\left(  \kappa_{m}^{3}I_{n}^{\prime}(\kappa_{m}a)+n^{2}\frac{1-\nu}{a^{2}
}\left(  -\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)+\frac{1}{a}I_{n}(\kappa
_{m}a)\right)  \right)
</math>
</center>
From the equations (11), (12) and the boundary conditions over the edges of the plate, 
we can write a linear system of equation, limiting the number of modes of the dispersion equation 
to <math>N</math> real ones
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
-A_0 K_n(k_0 a)&0 \quad \cdots&0\\
0&&\\
\vdots&-A_l K_n(k_l a)&\vdots\\
&&0\\
0&\cdots \quad 0 & -A_N K_n(k_N a)
\end{bmatrix}
&
\begin{bmatrix}
I_n(\kappa_{-2}a) B_{-20}&\cdots&I_n(\kappa_{N}a) B_{N0}\\
&&\\
\vdots&&\vdots\\
&&\\
I_n(\kappa_{-2}a) B_{-2N}&\cdots&I_n(\kappa_{N}a) B_{NN}
\end{bmatrix}
\\
\begin{bmatrix}
0&&\cdots&&0\\
0&&\cdots&&0
\end{bmatrix}
&
\begin{bmatrix}
L^1_{-2n}&\cdots&L^1_{Nn}\\
L^2_{-2n}&\cdots&L^2_{Nn}
\end{bmatrix}
\\
\begin{bmatrix}
-k_0 K'_n(k_0 a) A_0&0 \quad \cdots&0\\
0&&\\
\vdots&-k_l K'_n(k_l a) A_l&\vdots\\
&&0\\
0&\cdots \quad 0 & -k_N K'_n(k_N a) A_N
\end{bmatrix}
&
\begin{bmatrix}
\kappa_{-2} I'_n(\kappa_{-2}a) B_{-20}&\cdots&\kappa_{N} I'_n(\kappa_{N}a) B_{N0}\\
&&\\
\vdots&&\vdots\\
&&\\
\kappa_{-2} I'_n(\kappa_{-2}a) B_{-2N}&\cdots&\kappa_{N} I'_n(\kappa_{N}a) B_{NN}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0n} \\
\\
\vdots \\
\\
a_{Nn} \\
b_{-2n}\\
b_{-1n}\\
b_{0n}\\
\\
\vdots \\
\\
b_{Nn}
\end{bmatrix}
=
\begin{bmatrix}
\begin{bmatrix}
I_n(k_0 a) A_0&0 \quad \cdots&0\\
0&&\\
\vdots&I_n(k_l a) A_l&\vdots\\
&&0\\
0&\cdots \quad 0 & I_n(k_N a) A_N
\end{bmatrix}
\\
\begin{bmatrix}
0&&\cdots&&0\\
0&&\cdots&&0
\end{bmatrix}
\\
\begin{bmatrix}
k_0 I'_n(k_0 a) A_0&0 \quad \cdots&0\\
0&&\\
\vdots&k_l I'_n(k_l a) A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &k_N I'_n(k_N a) A_N
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
D_{0n} \\
\\
\vdots \\
\\
D_{Nn} \\
0\\
0\\
D_{0n}\\
\\
\vdots \\
\\
D_{Nn}
\end{bmatrix}
</math>
</center>
for each <math>n</math>.

Therefore we can find a [[Diffraction Transfer Matrix]] for each <math>n</math>, 
by setting
<center><math>
\forall i \in [0, N], (D_{pn})_{p \in [0, N]} = \delta_{ip}
</math></center>
Then we solve the linear system defined previously, so that we can find the coefficients
<math>(a_{ln})_{l \in [0, N]}</math> for each <math>i</math>.
This vector represents exactly the <math>i^{th}</math> column of the [[Diffraction Transfer Matrix]],
<math>n</math> being set.

This method permits to obtain the matrix which links the coefficients of the incident and scattered 
potential in the free water domain. Applying this for each <math>n</math>, we finally obtain a 3-dimensional
matrix for the [[Diffraction Transfer Matrix]].

== Matlab Code ==

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

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{elastic plate dispersion equation code}}

[[Category:Problems with Cylindrical Symmetry]]
[[Category:Floating Elastic Plate]]
[[Category:Eigenfunction Matching Method]]