Difference between revisions of "Eigenfunction Matching Method for Floating Elastic Plates"

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=Introduction=
+
{{pages to be deleted}}
 +
 
 +
==Introduction==
  
 
We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties
 
We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties
Line 10: Line 12:
 
the plate boundary conditions are satisfied as auxiliary equations.
 
the plate boundary conditions are satisfied as auxiliary equations.
  
= Equations =
+
== Equations ==
  
 
We consider the problem of small-amplitude waves which are incident on a set of floating elastic
 
We consider the problem of small-amplitude waves which are incident on a set of floating elastic
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<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
 
\left( \beta_\mu \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2  
 
\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 \;\;\;\;  
+
- \gamma_\mu\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0, \;\;\;\;  
 
\mbox{ at } z = 0, \;\;\; l_\mu \leq x \leq r_\mu,
 
\mbox{ at } z = 0, \;\;\; l_\mu \leq x \leq r_\mu,
 
\end{matrix}</math></center>
 
\end{matrix}</math></center>
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at the edges of the plates are
 
at the edges of the plates are
 
<center><math>\begin{matrix}
 
<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,
+
\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>
 
\end{matrix}</math></center>
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
\left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = l_\mu,r_\mu.
+
\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>
 
\end{matrix}</math></center>
 
where
 
where
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shown in Figure~35.
 
shown in Figure~35.
  
=Method of solution=
+
==Method of solution==
  
==Eigenfunction expansion==
+
===Eigenfunction expansion===
  
We will solve Eqs.~(13) to (17) using an [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]].
+
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]].
+
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.
 
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
 
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  
 
that extra modes are required in order to solve the higher order boundary  
 
conditions at the plate edges.   
 
conditions at the plate edges.   
The first and last plates are semi-infinite and the middle plates are finite.
+
 
 
The potential velocity of the first plate can be expressed as the summation of an incident wave and  
 
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  
 
of reflected waves, one of which is propagating but the rest of  
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the transmission and reflection are not expanded at opposite ends of the plate.   
 
the transmission and reflection are not expanded at opposite ends of the plate.   
  
===Separation of variables===
+
====Separation of variables====
 
The potential velocity can be written in terms of an infinite series of separated eigenfunctions under
 
The potential velocity can be written in terms of an infinite series of separated eigenfunctions under
 
each elastic plate, of the form  
 
each elastic plate, of the form  
<math></math>\phi = e^{\kappa_\mu x} \cos(k_\mu(z+h)).<math></math>
+
<math>\phi = e^{\kappa_\mu x} \cos(k_\mu(z+h)).</math>
If we apply the boundary conditions given by Eqs.~(14)
+
If we apply the boundary conditions given  
and (15)
 
 
we obtain the [[Dispersion Relation for a Floating Elastic Plate]]
 
we obtain the [[Dispersion Relation for a Floating Elastic Plate]]
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4}  + 1 - \alpha\gamma_\mu}
+
k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4}  + 1 - \alpha\gamma_\mu}  
(18)
 
 
\end{matrix}</math></center>
 
\end{matrix}</math></center>
Solving for <math>k_\mu</math>, this dispersion Eq.~\eqref{DispersionIce} gives a pure imaginary root
+
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, two complex roots (two complex conjugate paired roots
 
with positive imaginary part in all physical situations), an infinite number of positive real 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  
 
which approach <math>{n\pi}/{h}</math> as <math>n</math> approaches infinity, and also the negative of all  
these roots [[Fox and Squire 1994]] . We denote the two complex roots with positive imaginary part  
+
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  
 
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
 
root with positive imaginary part by <math>k_\mu(0)</math> and the real roots with positive imaginary part
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boundary and by applying the boundary conditions at the edge of each plate.
 
boundary and by applying the boundary conditions at the edge of each plate.
  
==Expressions for the potential velocity==
+
===Expressions for the potential velocity===
  
 
We now expand the potential under each plate using the separation of variables solution.  
 
We now expand the potential under each plate using the separation of variables solution.  
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at <math>M</math> real roots of the dispersion equation.  
 
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:
 
The potential <math>\phi</math> can now be expressed as the following sum of eigenfunctions:
<center><math>(19)
+
<center><math>
 
\phi \approx \left\{
 
\phi \approx \left\{
 
\begin{matrix}
 
\begin{matrix}
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<center><math>
 
<center><math>
 
\eta \approx \frac{i}{\omega}\left\{
 
\eta \approx \frac{i}{\omega}\left\{
\begin{matrix}{ll}
+
\begin{matrix}
 
{
 
{
 
Ik_1(0)e^{\kappa_{1}(0)(x-r_1)}\tan{(k_1(0)h)} - } \\
 
Ik_1(0)e^{\kappa_{1}(0)(x-r_1)}\tan{(k_1(0)h)} - } \\
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\end{matrix}\right.
 
\end{matrix}\right.
 
</math></center>
 
</math></center>
 
  
 
==Solving via eigenfunction matching==
 
==Solving via eigenfunction matching==
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Taking inner products leads to the following equations
 
Taking inner products leads to the following equations
<center><math>(20)
+
<center><math>
 
\begin{matrix}
 
\begin{matrix}
 
{
 
{
\int_{-h}^0 \phi_\mu(r_\mu,z)\cos \frac{m\pi}{h}(z+h) \, dz } &=&  
+
\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) \, dz }\\  
+
\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) \, dz } &=&  
+
\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) \, dz }
+
\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}
 
\end{matrix}
 
</math></center>
 
</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
 
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> given by Eq.~(19) valid for <math>l_\mu <x<r_\mu</math>.
+
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
 
The remaining equations to be solved are given by the two edge conditions satisfied at both
 
edges of each plate
 
edges of each plate
<center><math>(21)
+
<center><math>
 
\begin{matrix}
 
\begin{matrix}
 
{
 
{
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We will show the explicit form of the linear system of equations which arise  
 
We will show the explicit form of the linear system of equations which arise  
when we solve Eqs.~(20) and (21).  
+
when we solve these equations.  
 
Let <math>{\mathbf T}_\mu</math> be a column vector given by  
 
Let <math>{\mathbf T}_\mu</math> be a column vector given by  
 
<math>\left[T_{\mu}(-2), . . ., T_{\mu}(M)\right]^{{\mathbf T}}</math>  
 
<math>\left[T_{\mu}(-2), . . ., T_{\mu}(M)\right]^{{\mathbf T}}</math>  
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The equations which arise from matching at the boundary between the first
 
The equations which arise from matching at the boundary between the first
 
and second plate are
 
and second plate are
<center><math>(22)
+
<center><math>
 
\begin{matrix}
 
\begin{matrix}
 
I{\mathbf C} + {\mathbf M}^{+}_{R_1} {\mathbf R}_1 ={\mathbf M}^{-}_{T_2}  {\mathbf T}_2
 
I{\mathbf C} + {\mathbf M}^{+}_{R_1} {\mathbf R}_1 ={\mathbf M}^{-}_{T_2}  {\mathbf T}_2
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The equations which arise from matching at the boundary of the <math>\mu</math>th and (<math>\mu+1</math>)th plate
 
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  
 
boundary (<math>\mu>1</math>) are  
<center><math>(23)
+
<center><math>
 
\begin{matrix}
 
\begin{matrix}
 
{\mathbf M}^{+}_{T_\mu} {\mathbf T}_\mu +{\mathbf M}^{+}_{R_\mu} {\mathbf R}_\mu   
 
{\mathbf M}^{+}_{T_\mu} {\mathbf T}_\mu +{\mathbf M}^{+}_{R_\mu} {\mathbf R}_\mu   
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</math></center>
 
</math></center>
 
The equations which arise from matching at the (<math>\Lambda-1</math>)th and <math>\Lambda</math>th boundary are
 
The equations which arise from matching at the (<math>\Lambda-1</math>)th and <math>\Lambda</math>th boundary are
<center><math>(24)
+
<center><math>
 
\begin{matrix}
 
\begin{matrix}
 
{\mathbf M}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1}  
 
{\mathbf M}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1}  
Line 239: Line 240:
 
where <math>{\mathbf M}^{+}_{T_\mu}</math>, <math>{\mathbf M}^{+}_{R_\mu}</math>,
 
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
 
<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>(25)
+
<center><math>
 
\begin{matrix}
 
\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) \, dz}, \\  
+
{\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) \, dz },\\
+
{\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}^{-}_{T_\mu}(m,n) =  {\mathbf M}^{+}_{R_\mu}(m,n)  }\\
Line 260: Line 261:
 
</math></center>
 
</math></center>
 
<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by
 
<math>\mathbf{C}</math> is a <math>(M+1)</math> vector which is given by
<center><math>(26)
+
<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)\, dz.
+
{\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>
 
</math></center>
  
The integrals in Eqs.~(25) and (26) are each solved analytically. Now, for all but the first and <math>\Lambda</math>th plate, Eq.~(21) becomes
+
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>
 
<center><math>
 
\begin{matrix}
 
\begin{matrix}
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</math></center>
 
</math></center>
  
 
+
Now, the matching matrix is a <math>(2M+6)\times(\Lambda-1)</math> by  
\noindent Now, the matching matrix is a <math>(2M+6)\times(\Lambda-1)</math> by  
 
 
<math>(2M+1)\times(\Lambda -1)</math> matrix given by
 
<math>(2M+1)\times(\Lambda -1)</math> matrix given by
 
<center><math>
 
<center><math>
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</math></center>
 
</math></center>
  
\noindent the edge matrix is a <math>(2M+6)\times(\Lambda-1)</math> by <math>4(\Lambda-1)</math> matrix given by
+
the edge matrix is a <math>(2M+6)\times(\Lambda-1)</math> by <math>4(\Lambda-1)</math> matrix given by
  
 
<center><math>
 
<center><math>
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</math></center>
 
</math></center>
  
\noindent and finally the complete system to be solved is given by
+
and finally the complete system to be solved is given by
 
<center><math>
 
<center><math>
 
\left( \begin{matrix}
 
\left( \begin{matrix}
Line 376: Line 376:
 
The final system of equations has size <math>(2M+6)\times (\Lambda - 1)</math> by
 
The final system of equations has size <math>(2M+6)\times (\Lambda - 1)</math> by
 
<math>(2M+6)\times (\Lambda - 1)</math>.  
 
<math>(2M+6)\times (\Lambda - 1)</math>.  
The method of
 
solution we have derived is relatively simple and leads to large systems of equations when we simulate multiple plates. Our aim is to produce code which is simple to develop and which we have a strong degree of confidence is numerically accurate and error free. We do not want to make any kind of wide-spacing approximations since  real ice fields always have some small floes which we want to be able to simulate. We have used our method to solve for up to a hundred plates in simulations of wave propagation in the marginal ice zone.
 
 
The system of equations has a large
 
number of zero entries, due to the fact that each plate couples
 
only with its nearest neighbour. It seems likely that a more sophisticated method of solution could be developed, which exploits this structure.  We have have been unable to find such a method due to the difficulty of including the free edge conditions.
 
 
 
 
  
  
 
[[Category:Floating Elastic Plate]]
 
[[Category:Floating Elastic Plate]]
 
[[Category:Eigenfunction Matching Method]]
 
[[Category:Eigenfunction Matching Method]]

Latest revision as of 00:05, 17 October 2009


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]\displaystyle{ 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 Plates in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate)

[math]\displaystyle{ \begin{matrix} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} - k_y^2\right) \phi = 0, \;\;\;\; \mbox{ for } -h \lt z \leq 0, \end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix} \frac{\partial \phi}{\partial z} = 0, \;\;\;\; \mbox{ at } z = - h, \end{matrix} }[/math]
[math]\displaystyle{ \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]

where [math]\displaystyle{ \alpha = \omega^2 }[/math], [math]\displaystyle{ \beta_\mu }[/math] and [math]\displaystyle{ \gamma_\mu }[/math] and the stiffness and mass constant for the [math]\displaystyle{ \mu }[/math]th plate. The conditions at the edges of the plates are

[math]\displaystyle{ \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]
[math]\displaystyle{ \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]

where [math]\displaystyle{ l_\mu }[/math] and [math]\displaystyle{ r_\mu }[/math] represent the left and right edge of the [math]\displaystyle{ \mu }[/math]th plate as shown in Figure~35.

Method of solution

Eigenfunction expansion

We will solve the system of equations using an 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]\displaystyle{ x }[/math] tends to [math]\displaystyle{ -\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]\displaystyle{ +\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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \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

[math]\displaystyle{ \begin{matrix} k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4} + 1 - \alpha\gamma_\mu} \end{matrix} }[/math]

Solving for [math]\displaystyle{ 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]\displaystyle{ {n\pi}/{h} }[/math] as [math]\displaystyle{ 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]\displaystyle{ k_\mu(-2) }[/math] and [math]\displaystyle{ k_\mu(-1) }[/math], the purely imaginary root with positive imaginary part by [math]\displaystyle{ k_\mu(0) }[/math] and the real roots with positive imaginary part by [math]\displaystyle{ k_\mu(n) }[/math] for [math]\displaystyle{ n }[/math] a positive integer. The imaginary root with positive imaginary part corresponds to a reflected travelling mode propagating along the [math]\displaystyle{ 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]\displaystyle{ \kappa_\mu }[/math] is

[math]\displaystyle{ \kappa_\mu(n) = \sqrt{k_\mu(n)^2 + k_y^2}, }[/math]

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]\displaystyle{ 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]\displaystyle{ M }[/math] real roots of the dispersion equation. The potential [math]\displaystyle{ \phi }[/math] can now be expressed as the following sum of eigenfunctions:

[math]\displaystyle{ \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 \lt 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\lt x \lt 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\lt x, \end{matrix} \right. }[/math]

where [math]\displaystyle{ I }[/math] is the non-dimensional incident wave amplitude in potential, [math]\displaystyle{ \mu }[/math] is the [math]\displaystyle{ \mu^{th} }[/math] plate, [math]\displaystyle{ \Lambda }[/math] is the last plate, [math]\displaystyle{ r_\mu }[/math] represents the [math]\displaystyle{ x }[/math]-coordinate of the right edge of the [math]\displaystyle{ \mu^{th} }[/math] plate, [math]\displaystyle{ l_\mu }[/math] ([math]\displaystyle{ =r_{\mu-1} }[/math]) represents the [math]\displaystyle{ x }[/math]-coordinate of the left edge of the [math]\displaystyle{ \mu^{th} }[/math] plate, [math]\displaystyle{ R_\mu(n) }[/math] represents the reflected potential coefficient of the [math]\displaystyle{ n^{th} }[/math] mode under the [math]\displaystyle{ \mu^{th} }[/math] plate, and [math]\displaystyle{ T_\mu(n) }[/math] represents the transmitted potential coefficient of the [math]\displaystyle{ n^{th} }[/math] mode under the [math]\displaystyle{ \mu^{th} }[/math] plate. Note that we have divided by [math]\displaystyle{ \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

[math]\displaystyle{ \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 \lt 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\lt x \lt 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\lt x. \end{matrix}\right. }[/math]

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]\displaystyle{ x }[/math]-direction at each plate boundary. We impose the latter condition by taking inner products with respect to the orthogonal functions [math]\displaystyle{ \cos \frac{m\pi}{h}(z+h) }[/math], where [math]\displaystyle{ m }[/math] is a natural number. These functions are chosen for the following reasons. The vertical eigenfunctions [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \cos{(m\pi/h)(z + h)} }[/math] for large [math]\displaystyle{ 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

[math]\displaystyle{ \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]

where [math]\displaystyle{ m\in[0,M] }[/math] and [math]\displaystyle{ \phi_\mu }[/math] denotes the potential under the [math]\displaystyle{ \mu }[/math]th plate, i.e. the expression for [math]\displaystyle{ \phi }[/math] valid for [math]\displaystyle{ l_\mu \lt x\lt r_\mu }[/math]. The remaining equations to be solved are given by the two edge conditions satisfied at both edges of each plate

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \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]

The equations which arise from matching at the boundary of the [math]\displaystyle{ \mu }[/math]th and ([math]\displaystyle{ \mu+1 }[/math])th plate boundary ([math]\displaystyle{ \mu\gt 1 }[/math]) are

[math]\displaystyle{ \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]

The equations which arise from matching at the ([math]\displaystyle{ \Lambda-1 }[/math])th and [math]\displaystyle{ \Lambda }[/math]th boundary are

[math]\displaystyle{ \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]

where [math]\displaystyle{ {\mathbf M}^{+}_{T_\mu} }[/math], [math]\displaystyle{ {\mathbf M}^{+}_{R_\mu} }[/math], [math]\displaystyle{ {\mathbf M}^{-}_{T_\mu} }[/math], and [math]\displaystyle{ {\mathbf M}^{-}_{R_\mu} }[/math]are [math]\displaystyle{ (M+1) }[/math] by [math]\displaystyle{ (M+3) }[/math] matrices given by

[math]\displaystyle{ \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]

[math]\displaystyle{ {\mathbf N}^{+}_{T_\mu} }[/math], [math]\displaystyle{ {\mathbf N}^{+}_{R_\mu} }[/math], [math]\displaystyle{ {\mathbf N}^{-}_{T_\mu} }[/math], and [math]\displaystyle{ {\mathbf N}^{-}_{R_\mu} }[/math] are given by

[math]\displaystyle{ \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]

[math]\displaystyle{ \mathbf{C} }[/math] is a [math]\displaystyle{ (M+1) }[/math] vector which is given by

[math]\displaystyle{ {\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]

The integrals in the above equation are each solved analytically. Now, for all but the first and [math]\displaystyle{ \Lambda }[/math]th plate, the edge equation becomes

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \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]

and for the [math]\displaystyle{ \Lambda }[/math]th plate we have no reflection so

[math]\displaystyle{ \begin{matrix} {\mathbf E}^{-}_{T_\mu} {\mathbf T}_\mu = 0.\\ \end{matrix} }[/math]

[math]\displaystyle{ {\mathbf E}^{+}_{T_\mu} }[/math], [math]\displaystyle{ {\mathbf E}^{+}_{R_\mu} }[/math], [math]\displaystyle{ {\mathbf E}^{-}_{T_\mu} }[/math] and [math]\displaystyle{ {\mathbf E}^{-}_{R_\mu} }[/math] are 2 by M+3 matrices given by

[math]\displaystyle{ \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]

Now, the matching matrix is a [math]\displaystyle{ (2M+6)\times(\Lambda-1) }[/math] by [math]\displaystyle{ (2M+1)\times(\Lambda -1) }[/math] matrix given by

[math]\displaystyle{ {\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]

the edge matrix is a [math]\displaystyle{ (2M+6)\times(\Lambda-1) }[/math] by [math]\displaystyle{ 4(\Lambda-1) }[/math] matrix given by

[math]\displaystyle{ {\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]

and finally the complete system to be solved is given by

[math]\displaystyle{ \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]

The final system of equations has size [math]\displaystyle{ (2M+6)\times (\Lambda - 1) }[/math] by [math]\displaystyle{ (2M+6)\times (\Lambda - 1) }[/math].