View source for Eigenfunction Matching for a Finite Floating Elastic Plate using Symmetry

{{complete pages}}

==Introduction==

We show here a solution for a finite [[:Category:Floating Elastic Plate|Floating Elastic Plate]] on [[Finite Depth]]. 
The simpler solution for a [[Eigenfunction Matching for a Semi-Infinite Floating Elastic Plate|Semi-Infinite Elastic Plate]] describes
many of the ideas here in more detail. We use [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]].

== Equations ==

{{finite plate frequency domain}}

==Method of solution==

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

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a floating elastic plate}}

{{free surface floating elastic plate relations}}

==Symmetric solution==

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>x=0</math> have to be equal.
We also truncate the sum at <math>N</math> being careful that we have
two extra modes on the plate covered region to satisfy the edge conditions. 
We use [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]]
and we express the potential as 
<center>
<math>
\phi = e^{-k_0 (x+L)}\phi_{0}\left(  z\right) + \sum_{m=0}^{N}
a_{m}^{s} e^{k_n (x+L)}\phi_{m}\left(  z\right),\,\,x<-L
</math>
</center>
<center>
<math>
\phi  =\sum_{m=-2}^{N}b_{m}^{s}\frac{\cosh(\kappa_n x)}{\cosh(\kappa_n L)}\psi_{m}(z),\,\,-L\leq x\leq 0
</math>
</center>
When we equate the potential and its derivative at <math>x=-L</math> we obtain 
<center>
<math>
\phi_{0}\left(  z\right) + \sum_{m=0}^{N}
a_{m}^{s} \phi_{m}\left(  z\right) 
=\sum_{m=-2}^{N}b_{m}^{s}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
 -k_{0}\phi_{0}\left(  z\right)  +\sum
_{m=0}^{N} a_{m}^{s}k_{m}\phi_{m}\left(  z\right) 
 =-\sum_{m=-2}^{N}b_{m}^{s}\kappa_{m}\tanh(\kappa_m h) \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>
A_{0}\delta_{0l}+a_{l}^{s}A_{l}
=\sum_{m=-2}^{N}b_{m}^{s}B_{ml},\,\,0 \leq l  \leq N
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}^{s}k_{l}A_l
 =-\sum_{m=-2}^{N}b_{m}^{s}\kappa_{m}\tanh(\kappa_m L) B_{ml},\,\,0 \leq l \leq N
</math>
</center>
If we multiply the first equation by <math>k_l</math> and subtract the second equation
we obtain 
<center>
<math>
2k_{0}A_{0}\delta_{0l}
 =\sum_{m=-2}^{N}b_{m}^{s}(k_l + \tanh(\kappa_m L)\kappa_{m})B_{ml}
</math>
</center>
Finally, we need to apply the conditions at the edge of the plate to give us two further equations,
<center>
<math>
\partial_x^2\partial_z\phi = - \sum_{m=-2}^{N}b_{m}^{s} \kappa_m^3 \tan\kappa_m h = 0
</math>
</center>
and
<center>
<math>
\partial_x^3\partial_z\phi = \sum_{m=-2}^{N}b_{m}^{s} \kappa_m^4 \tanh(\kappa_m L)\tan\kappa_m h = 0
</math>
</center>

=== Waves Incident at an Angle ===

We can consider the problem when the waves are incident at an angle <math>\theta</math>. 
{{incident angle}}

This means that the potential is now of the form <math>\phi(x,y,z)=e^{k_y y}\phi(x,z)</math>
so that when we separate variables we obtain 

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}^{s}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<-L
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=-2}^{\infty}b_{m}^{s}
\frac{\cosh(\hat{\kappa}_{m}x)}{\cosh(\hat{\kappa}_{m} L)} \psi_{m}(z), \;\;-L \leq x \leq 0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 + k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 + k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part. 

The edge conditions are also different and are
<center><math>\begin{matrix}
\left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0,  
\end{matrix}</math></center>
and
<center><math>\begin{matrix}
\left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0,
\end{matrix}</math></center>
where <math>\nu</math> is Poisons ratio. 

We can expend these edge conditions, which respectively gives 
<center>
<math>
-\sum_{m=-2}^{\infty}b_{m}\kappa_m (\hat{\kappa}_m^3-\hat{\kappa}_m k_y^2(2-\nu))\tanh(\hat{\kappa}_m L)\tan \kappa_m h=0
</math>
</center>
and
<center>
<math>
-\sum_{m=-2}^{\infty}b_{m}\kappa_m (\hat{\kappa}_m^2-k_y^2\nu))\tan \kappa_m h=0
</math>
</center>

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}^{s}A_{l}
=\sum_{m=-2}^{\infty}b_{m}^{s}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}^{s}\hat{k}_{l}A_l
 =-\sum_{m=-2}^{\infty}b_{m}^{s}\hat{\kappa}_{m}\tanh(\kappa_m L)B_{ml} 
</math>
</center>
and these are solved exactly as before.

==Anti-symmetric solution==

The anti-symmetric potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0 x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}^{a}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<-L
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=-2}^{\infty}b_{m}^{a}
\frac{\sinh(\hat{\kappa}_{m} x)}{\sinh(-\hat{\kappa}_{m} L)} \psi_{m}(z), \;\;-L \leq x \leq 0
</math>
</center>

The edge conditions are
<center>
<math>
\sum_{m=-2}^{\infty}b_{m}^{a}\kappa_m (\hat{\kappa}_m^3-\hat{\kappa}_m k_y^2(2-\nu))\coth(\hat{\kappa}_m L)\tan \kappa_m h=0
</math>
</center>
and
<center>
<math>
-\sum_{m=-2}^{\infty}b_{m}^{a}\kappa_m (\hat{\kappa}_m^2-k_y^2\nu))\tan \kappa_m h=0
</math>
</center>

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}^{a}A_{l}
=\sum_{m=-2}^{\infty}b_{m}^{a}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}^{a}\hat{k}_{l}A_l
 =-\sum_{m=-2}^{\infty}b_{m}^{a}\hat{\kappa}_{m}\coth(\kappa_m L)B_{ml} 
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/finite_plate_symmetry.m finite_plate_symmetry.m]

=== Additional code ===

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

[[Category:Floating Elastic Plate]]
[[Category:Eigenfunction Matching Method]]
[[Category:Symmetry in Two Dimensions]]
[[Category:Symmetry in Two Dimensions]]
[[Category:Complete Pages]]