View source for Eigenfunction Matching Method for Floating Elastic Plates

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==Introduction==

We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties
This work is based on [[Kohout et. al. 2006]]. This is a generalisation of the 
[[Eigenfunction Matching Method for a Semi-Infinite Floating Elastic Plate]].
We assume that the first and last plate are semi-infinite. The presentation here does not
allow open water (it could be included but makes the formulation more complicated). In any case
open water can be considered by taking the limit as the
plate thickness tends to zero. The solution is derived using an extended eigenfunction matching method, in which
the plate boundary conditions are satisfied as auxiliary equations.

== Equations ==

We consider the problem of small-amplitude waves which are incident on a set of floating elastic
plates occupying the entire water surface. The submergence of the plates is considered negligible. 
We assume that the problem is invariant in the <math>y</math> direction, although we allow the waves to be
incident from an angle. 
The set of plates consists of two semi-infinite plates, separated by a region which
consists of a finite number of plates with variable properties. 
We also assume that the plate edges are free to move at
each boundary, although other boundary conditions could easily be considered using
the methods of solution presented here. We begin with the [[Frequency Domain Problem]] for multiple
[[Floating Elastic Plate|Floating Elastic Plates]]
in the non-dimensional form of [[Tayler_1986a|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]])

<center><math>\begin{matrix}
\left(\frac{\partial^2}{\partial x^2} + 
\frac{\partial^2}{\partial z^2} - k_y^2\right) \phi = 0, \;\;\;\; \mbox{ for } -h < z \leq 0, 
\end{matrix}</math></center>
<center><math>\begin{matrix}
\frac{\partial \phi}{\partial z} = 0, \;\;\;\; \mbox{ at } z = - h,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\left( \beta_\mu \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 
- \gamma_\mu\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0, \;\;\;\; 
\mbox{ at } z = 0, \;\;\; l_\mu \leq x \leq r_\mu,
\end{matrix}</math></center>
where <math>\alpha = \omega^2</math>, <math>\beta_\mu</math> and <math>\gamma_\mu</math>
and the stiffness and mass constant for the <math>\mu</math>th plate. The conditions
at the edges of the plates 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, \;\;\;\; \mbox{ at } z = 0, \;\;\; \mbox{ for } x = l_\mu,r_\mu,
\end{matrix}</math></center>
<center><math>\begin{matrix}
\left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0,
\;\;\;\;\mbox{ at } z = 0, \;\;\; \mbox{ for } x = l_\mu,r_\mu.
\end{matrix}</math></center>
where
<math>l_\mu</math> and <math>r_\mu</math> represent the left and right edge of the <math>\mu</math>th plate as
shown in Figure~35.

==Method of solution==

===Eigenfunction expansion===

We will solve the system of equations using an [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]].
The method was developed by [[Fox and Squire 1994]] for the case of a single plate
as the research is described in [[Two-Dimensional Floating Elastic Plate]].
We show here how this method can be extended to the case of an arbitrary number of plates.
One of the key features in the eigenfunction expansion method for elastic plates is
that extra modes are required in order to solve the higher order boundary 
conditions at the plate edges.  

The potential velocity of the first plate can be expressed as the summation of an incident wave and 
of reflected waves, one of which is propagating but the rest of 
which are evanescent and they decay as <math>x</math> tends to <math>-\infty</math>.
Similarly the potential under the final plate can be expressed as a sum of transmitting waves,
one of which is propagating and the rest of which are evanescent and decay towards <math>+\infty</math>.
The potential under the middle plates can be expressed as the sum of transmitting waves and reflected waves,
each of which consists of a propagating wave plus evanescent waves which decay as <math>x</math> decreases or
increases respectively. We could combine these waves in the formulation, but because of the exponential
growth (or decay) in the <math>x</math> direction the solution becomes numerically unstable in some cases if
the transmission and reflection are not expanded at opposite ends of the plate.  

====Separation of variables====
The potential velocity can be written in terms of an infinite series of separated eigenfunctions under
each elastic plate, of the form 
<math>\phi = e^{\kappa_\mu x} \cos(k_\mu(z+h)).</math>
If we apply the boundary conditions given 
we obtain the [[Dispersion Relation for a Floating Elastic Plate]]
<center><math>\begin{matrix}
k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4}  + 1 - \alpha\gamma_\mu} 
\end{matrix}</math></center>
Solving for <math>k_\mu</math> gives a pure imaginary root
with positive imaginary part, two complex roots (two complex conjugate paired roots
with positive imaginary part in all physical situations), an infinite number of positive real roots 
which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all 
these roots ([[Dispersion Relation for a Floating Elastic Plate]]) . We denote the two complex roots with positive imaginary part 
by <math>k_\mu(-2)</math> and <math>k_\mu(-1)</math>, the purely imaginary 
root with positive imaginary part by <math>k_\mu(0)</math> and the real roots with positive imaginary part
by <math>k_\mu(n)</math> for <math>n</math> a positive integer. 
The imaginary root with positive imaginary part corresponds to a 
reflected travelling mode propagating along the <math>x</math> axis. 
The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes.
In a similar manner, the negative of these correspond to the transmitted travelling, damped and evanescent modes respectively.
The coefficient <math>\kappa_\mu</math> is 
<center><math>\kappa_\mu(n) = \sqrt{k_\mu(n)^2 + k_y^2},</math></center>
where the root with positive real part is chosen or if the real part is negative with negative imaginary part. 
Note that the solutions of the dispersion equation will be different under plates of different
properties, and that the expansion is only valid under a single plate. We will solve for the 
coefficients in the expansion by matching the potential and its <math>x</math> derivative at each 
boundary and by applying the boundary conditions at the edge of each plate.

===Expressions for the potential velocity===

We now expand the potential under each plate using the separation of variables solution. 
We always include the two complex and one imaginary root, and truncate the expansion
at <math>M</math> real roots of the dispersion equation. 
The potential <math>\phi</math> can now be expressed as the following sum of eigenfunctions:
<center><math>
\phi \approx \left\{
\begin{matrix}
{
Ie^{\kappa_{1}(0)(x-r_1)}\frac{\cos(k_1(0)(z+h))}{\cos(k_1(0)h)} }+\\
 { 
        \qquad \qquad \sum_{n=-2}^{M}R_1(n) e^{\kappa_{1}(n)(x-r_1)} \frac{\cos(k_1(n)(z+h))}{\cos(k_1(n)h)} },& 
                \mbox{ for } x < r_1,\\
{
\sum_{n=-2}^{M}T_{\mu}(n) e^{-\kappa_\mu(n)(x-l_\mu)} 
        \frac{\cos(k_\mu(n)(z+h))}{\cos(k_\mu(n)h)} } + \\
{
                \qquad \qquad \sum_{n=-2}^{M}R_{\mu}(n) e^{\kappa_\mu(n)(x-r_\mu)} 
                        \frac{\cos(k_\mu(n)(z+h))}{\cos(k_\mu(n)h)} },
                                &\mbox{ for } l_\mu< x < r_\mu,\\
{
\sum_{n=-2}^{M}T_{\Lambda}(n)e^{-\kappa_{\Lambda}(n)(x-l_\Lambda)} 
        \frac{\cos(k_\Lambda(n)(z+h))}{\cos(k_\Lambda(n)h)} }, &\mbox{ for } l_\Lambda<x,
\end{matrix} \right.
</math></center>
where <math>I</math> is the non-dimensional incident wave amplitude in potential,
<math>\mu</math> is the <math>\mu^{th}</math> plate, <math>\Lambda</math> is the last plate,
<math>r_\mu</math> represents the <math>x</math>-coordinate of the right edge of the <math>\mu^{th}</math> plate,
<math>l_\mu</math> (<math>=r_{\mu-1}</math>) represents the <math>x</math>-coordinate of the left edge of the <math>\mu^{th}</math> plate,
<math>R_\mu(n)</math> represents the reflected potential coefficient of the <math>n^{th}</math> mode under the <math>\mu^{th}</math> plate, and 
<math>T_\mu(n)</math> represents the transmitted potential coefficient of the <math>n^{th}</math> mode under the <math>\mu^{th}</math> plate.
Note that we have divided by  <math>\cos{(kh)}</math>, 
so that the coefficients are normalised by the
potential at the free surface rather than at the bottom surface.

==Expressions for displacement==
The displacement is given by 
<center><math>
\eta \approx \frac{i}{\omega}\left\{
\begin{matrix}
{
Ik_1(0)e^{\kappa_{1}(0)(x-r_1)}\tan{(k_1(0)h)} - } \\
{
        \qquad \sum_{n=-2}^{M}R_{1}(n)k_1(n)e^{\kappa_{1}(n)(x-r_1)} \tan{(k_1(n)h)} }, 
                & \mbox{ for } x < r_1, \\
{
-\sum_{n=-2}^{M}T_{\mu}(n)k_\mu(n)e^{-\kappa_\mu(n)(x-l_\mu)}\tan{(k_\mu(n)h)} - }\\
{
        \qquad \sum_{n=-2}^{M}R_{\mu}(n)k_\mu(n)e^{\kappa_\mu(n)(x-r_\mu)} 
                \tan{(k_\mu(n)h)} },& \mbox{ for } l_\mu< x < r_\mu,\\
{
-\sum_{n=-2}^{M}T_{\Lambda}(n)k_\mu(n)e^{-\kappa_\mu(n)(x-l_\Lambda)}\tan{(k_\mu(n)h)} },
        &\mbox{ for } l_\Lambda<x.
\end{matrix}\right.
</math></center>

==Solving via eigenfunction matching==

To solve for the coefficients, we require as many equations as we have unknowns. 
We derive the equations from the free edge conditions and from imposing conditions of
continuity of the potential and its derivative in the <math>x</math>-direction at each plate
boundary. We impose the latter condition by taking inner products with respect to
the orthogonal functions <math>\cos \frac{m\pi}{h}(z+h)</math>, where <math>m</math> is a natural number. These functions are chosen for the following reasons.
The vertical eigenfunctions <math>\cos k_\mu(n)(z+h)</math> are not orthogonal (they are not even a basis) and could therefore lead to an ill-conditioned system of equations. Furthermore, by choosing  <math>\cos \frac{m\pi}{h}(z+h)</math> we can use the same functions to take the inner products under every plate. Finally, and most importantly, the plate eigenfunctions approach <math>\cos{(m\pi/h)(z + h)}</math> for large <math>m</math>, so that as we increase the number of modes the matrices become almost diagonal, leading to a very well-conditioned system of equations. 

Taking inner products leads to the following equations
<center><math>
\begin{matrix}
{
\int_{-h}^0 \phi_\mu(r_\mu,z)\cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } &=& 
{
\int_{-h}^0 \phi_{\mu+1}(l_{\mu+1},z)\cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z }\\ 
{
\int_{-h}^0 \frac{\partial\phi_\mu}{\partial x}(r_\mu,z) \cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } &=& 
{
\int_{-h}^0 \frac{\partial\phi_{\mu+1}}{\partial x}(l_{\mu+1},z) \cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z }
\end{matrix}
</math></center>
where <math>m\in[0,M]</math> and <math>\phi_\mu</math> denotes the potential under the <math>\mu</math>th plate, i.e. the expression
for <math>\phi</math> valid for <math>l_\mu <x<r_\mu</math>.
The remaining equations to be solved are given by the two edge conditions satisfied at both
edges of each plate
<center><math>
\begin{matrix}
{
\left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k_y^2\frac{\partial}{\partial x}\right)\frac{\partial\phi_\mu}{\partial z} }
&=&0, & \mbox{ for } z = 0 \mbox{ and } x = l_\mu,r_\mu,\\
{
\left(\frac{\partial^2}{\partial x^2} - \nu k_y^2\right)\frac{\partial\phi_\mu}{\partial z}  }
&=&0, & \mbox{ for } z = 0 \mbox{ and } x = l_\mu,r_\mu.
\end{matrix} 
</math></center>

We will show the explicit form of the linear system of equations which arise 
when we solve these equations. 
Let <math>{\mathbf T}_\mu</math> be a column vector given by 
<math>\left[T_{\mu}(-2), . . ., T_{\mu}(M)\right]^{{\mathbf T}}</math> 
and <math>{\mathbf R}_\mu</math> be a column vector given by  
<math>\left[R_{\mu}(-2) . . . R_{\mu}(M)\right]^{{\mathbf T}}</math>.

The equations which arise from matching at the boundary between the first
and second plate are
<center><math>
\begin{matrix}
I{\mathbf C} + {\mathbf M}^{+}_{R_1} {\mathbf R}_1 ={\mathbf M}^{-}_{T_2}  {\mathbf T}_2
+ {\mathbf M}^{-}_{R_2} {\mathbf R}_2,\\
-\kappa_1(0)I\mathbf{C} + {\mathbf N}^{+}_{R_1} {\mathbf R}_1 = {\mathbf N}^{-}_{T_2} {\mathbf T}_2
+ {\mathbf N}^{-}_{R_2} {\mathbf R}_2.
\end{matrix}
</math></center>
The equations which arise from matching at the boundary of the <math>\mu</math>th and (<math>\mu+1</math>)th plate
boundary (<math>\mu>1</math>) are 
<center><math>
\begin{matrix}
{\mathbf M}^{+}_{T_\mu} {\mathbf T}_\mu +{\mathbf M}^{+}_{R_\mu} {\mathbf R}_\mu  
={\mathbf M}^{-}_{T_{\mu+1}} {\mathbf T}_{\mu+1} 
 + {\mathbf M}^{-}_{ R_{\mu+1}} {\mathbf R}_{\mu+1}, \\
{\mathbf N}^{+}_{T_\mu} {\mathbf T}_\mu + {\mathbf N}^{+}_{R_\mu}  {\mathbf R}_\mu 
={\mathbf N}^{-}_{T_{\mu+1}}  {\mathbf T}_{\mu +1} 
+{\mathbf N}^{-}_{ R_{\mu +1}}  {\mathbf R}_{\mu +1}.
\end{matrix}
</math></center>
The equations which arise from matching at the (<math>\Lambda-1</math>)th and <math>\Lambda</math>th boundary are
<center><math>
\begin{matrix}
{\mathbf M}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1} 
+ {\mathbf M}^{+}_{R_{\Lambda-1}}  {\mathbf R}_{\Lambda-1} 
=  {\mathbf M}^{-}_{T_\Lambda } {\mathbf T}_{\Lambda}, \\
 {\mathbf N}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1}
 + {\mathbf N}^{+}_{R_{\Lambda-1}} {\mathbf R}_{\Lambda-1} =  {\mathbf N}^{-}_{T_\Lambda } {\mathbf T}_{\Lambda},
\end{matrix}
</math></center>

where <math>{\mathbf M}^{+}_{T_\mu}</math>, <math>{\mathbf M}^{+}_{R_\mu}</math>,
<math>{\mathbf M}^{-}_{T_\mu}</math>, and <math>{\mathbf M}^{-}_{R_\mu}</math>are <math>(M+1)</math> by <math>(M+3)</math> matrices given by
<center><math>
\begin{matrix}
{
{\mathbf M}^{+}_{T_\mu}(m,n) = \int_{-h}^0 e^{-\kappa_\mu(n) (r_\mu-l_\mu )} \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, \mathrm{d}z}, \\ 
{
{\mathbf M}^{+}_{R_\mu}(m,n) = \int_{-h}^0 \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, \mathrm{d}z },\\
{
{\mathbf M}^{-}_{T_\mu}(m,n) =  {\mathbf M}^{+}_{R_\mu}(m,n)  }\\
{
{\mathbf M}^{-}_{R_\mu}(m,n) = {\mathbf M}^{+}_{T_\mu}(m,n).  }
\end{matrix}
</math></center>
<math>{\mathbf N}^{+}_{T_\mu}</math>, <math>{\mathbf N}^{+}_{R_\mu}</math>,
<math>{\mathbf N}^{-}_{T_\mu}</math>, and <math>{\mathbf N}^{-}_{R_\mu}</math> are given by
<center><math>
\begin{matrix}
{\mathbf N}^{\pm}_{T_\mu}(m,n)= -\kappa_\mu(n){\mathbf M}^{\pm}_{T_\mu}(m,n),\\
{\mathbf N}^{\pm}_{R_\mu}(m,n)= \kappa_\mu(n){\mathbf M}^{\pm}_{R_\mu}(m,n).
\end{matrix}
</math></center>
<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by
<center><math>
{\mathbf C}(m)=\int_{-h}^0 \frac{\cos (k_1(0)(z+h))}{\cos (k_1(0)h)} \cos \left(\frac{m\pi}{h}(z+h)\right)\, \mathrm{d}z.
</math></center>

The integrals in the above equation are each solved analytically. Now, for all but the first and <math>\Lambda</math>th plate, the edge equation becomes
<center><math>
\begin{matrix}
{\mathbf E}^{+}_{T_\mu} {\mathbf T}_\mu + {\mathbf E}^{+}_{R_\mu} {\mathbf R}_\mu = 0,\\
{\mathbf E}^{-}_{T_\mu} {\mathbf T}_\mu + {\mathbf E}^{-}_{R_\mu} {\mathbf R}_\mu = 0.
\end{matrix}
</math></center>
The first and last plates only require two equations, because each has only one plate
edge. 
The equation for the first plate must be modified to include the effect of
the incident wave. This gives us 
<center><math>
\begin{matrix}
I \left(
\begin{matrix}
{\mathbf E}^{+}_{T_1}(1,0)\\ 
{\mathbf E}^{+}_{T_1}(2,0)
\end{matrix}
\right)
+ {\mathbf E}^{+}_{R_1} {\mathbf R}_1 = 0,\\
\end{matrix}
</math></center>
and for the <math>\Lambda</math>th plate we have no reflection so
<center><math>
\begin{matrix}
{\mathbf E}^{-}_{T_\mu} {\mathbf T}_\mu = 0.\\
\end{matrix}
</math></center>

<math>{\mathbf E}^{+}_{T_\mu}</math>, <math>{\mathbf E}^{+}_{R_\mu}</math>, <math>{\mathbf E}^{-}_{T_\mu}</math> and <math>{\mathbf E}^{-}_{R_\mu}</math>  are 2 by M+3 matrices given by 
<center><math>
\begin{matrix}
{\mathbf E}^{-}_{T_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(k_{\mu}(n)\kappa_\mu(n)\tan{(k_{\mu}(n)h)}),\\
{\mathbf E}^{+}_{T_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(k_{\mu}(n)\kappa_\mu(n)e^{-\kappa_\mu(n)(r_\mu - l_\mu)}\tan{(k_{\mu}(n)h)}),\\
{\mathbf E}^{-}_{R_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(-k_{\mu}(n)\kappa_\mu(n)e^{\kappa_\mu(n)(l_\mu - r_\mu)}\tan{(k_{\mu}(n)h)}),\\
{\mathbf E}^{+}_{R_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(-k_{\mu}(n)\kappa_\mu(n)\tan{(k_{\mu}(n)h)}),\\

{\mathbf E}^{-}_{T_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)\tan{(k_{\mu}(n)h)}),\\
{\mathbf E}^{+}_{T_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)e^{-\kappa_\mu(n)(r_\mu - l_\mu)}\tan{(k_{\mu}(n)h)}),\\
{\mathbf E}^{-}_{R_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)e^{\kappa_\mu(n)(l_\mu - r_\mu)}\tan{(k_{\mu}(n)h)}),\\
{\mathbf E}^{+}_{R_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)\tan{(k_{\mu}(n)h)}).\\
\end{matrix}
</math></center>

Now, the matching matrix is a <math>(2M+6)\times(\Lambda-1)</math> by 
<math>(2M+1)\times(\Lambda -1)</math> matrix given by
<center><math>
{\mathbf M} = 
\left( \begin{matrix}
{\mathbf M}^{+}_{R_1} & -{\mathbf M}^{-}_{T_2} & -{\mathbf M}^{-}_{R_2} &     0    &     0    &        & 0 & 0 & 0 \\ 
{\mathbf N}^{+}_{R_1} & -{\mathbf N}^{-}_{T_2} & -{\mathbf N}^{-}_{R_2} &     0    &     0    &        & 0 & 0 & 0 \\ 
   0    &  {\mathbf M}^{+}_{T_2} &  {\mathbf M}^{+}_{R_2} & -{\mathbf M}^{-}_{T_3} & -{\mathbf M}^{-}_{R_3} &  & 0 & 0 & 0 \\ 
   0    &  {\mathbf N}^{+}_{T_2} &  {\mathbf N}^{+}_{R_2} & -{\mathbf N}^{-}_{T_3} & -{\mathbf N}^{-}_{R_3} &        & 0 & 0 & 0 \\ 
        &          &  \vdots  &          &          & \ddots & \\
   0    &     0    &    0     &     0    &     0    &        & {\mathbf M}^{+}_{T_{\Lambda - 1}} & {\mathbf M}^{+}_{R_{\Lambda - 1}} & -{\mathbf M}^{-}_{ T_{\Lambda}} \\ 
   0    &     0    &    0     &     0    &     0    &        & {\mathbf N}^{+}_{T_{\Lambda - 1}} & {\mathbf N}^{+}_{R_{\Lambda - 1}} & -{\mathbf N}^{-}_{T_{\Lambda }} \\ 
\end{matrix} \right),
</math></center>

the edge matrix is a <math>(2M+6)\times(\Lambda-1)</math> by <math>4(\Lambda-1)</math> matrix given by

<center><math>
{\mathbf E} = 
\left( \begin{matrix}
  {\mathbf E}^{+}_{R_1} &    0   &           0        &     0    &     0    &        & 0 & 0 & 0 \\
   0    &  {\mathbf E}^{+}_{T_2} &  {\mathbf E}^{+}_{R_2} &     0    &     0    &        & 0 & 0 & 0 \\ 
   0    &  {\mathbf E}^{-}_{T_2} &  {\mathbf E}^{-}_{R_2} &     0    &     0    &        & 0 & 0 & 0 \\ 
   0    &     0    &     0    &  {\mathbf E}^{+}_{T_3} &  {\mathbf E}^{+}_{R_3} &  & 0 & 0 & 0 \\ 
   0    &     0    &     0    &  {\mathbf E}^{-}_{T_3} &  {\mathbf E}^{-}_{R_3} &        & 0 & 0 & 0 \\ 
        &          &  \vdots  &          &          & \ddots & \\ 
   0    &     0    &    0     &     0    &     0    &        & {\mathbf E}^{+}_{T_{\Lambda-1}} &  {\mathbf E}^{+}_{R_{\Lambda-1}} & 0 \\
   0    &     0    &    0     &     0    &     0    &        & {\mathbf E}^{-}_{T_{\Lambda-1}} &  {\mathbf E}^{-}_{R_{\Lambda-1}} & 0 \\
   0    &     0    &    0     &     0    &     0    &        & 0 & 0 & {\mathbf E}^{-}_{ T_\Lambda}
\end{matrix} \right),
</math></center>

and finally the complete system to be solved is given by
<center><math>
\left( \begin{matrix}
{\mathbf M}\\
{\mathbf E}\\
\end{matrix} \right)
\times
\left( \begin{matrix}
{\mathbf R}_1\\
{\mathbf T}_2\\
{\mathbf R}_2\\
{\mathbf T}_3\\
{\mathbf R}_3\\
\vdots\\
{\mathbf T}_{\Lambda-1}\\
{\mathbf R}_{\Lambda-1}\\
{\mathbf T}_{\Lambda}

\end{matrix} \right)
=
\left( \begin{matrix}
-I{\mathbf C}\\
\kappa_{1}(0)I{\mathbf C}\\
0\\
\vdots\\
-IE^{+}_{T_1}(1,0)\\ 
-IE^{+}_{T_1}(2,0)\\
0\\
\vdots

\end{matrix} \right).
</math></center>

The final system of equations has size <math>(2M+6)\times (\Lambda - 1)</math> by
<math>(2M+6)\times (\Lambda - 1)</math>. 


[[Category:Floating Elastic Plate]]
[[Category:Eigenfunction Matching Method]]