Difference between revisions of "Meylan 2002b"

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pp 387-402, 2002.
 
pp 387-402, 2002.
  
[[Time-Dependent Linear Water Wave]] problem of a [[Floating Elastic Plate]]
+
[[:Category:Time-Dependent Linear Water Waves|Time-Dependent Linear Water Wave]] problem of a [[Floating Elastic Plate]]
 
on [[Shallow Depth]]. The solution was found using a [[Generalised Eigenfunction Expansion]]
 
on [[Shallow Depth]]. The solution was found using a [[Generalised Eigenfunction Expansion]]
 
and as a sum over [[Scattering Frequencies]].
 
and as a sum over [[Scattering Frequencies]].
  
 
[[Category:Reference]]
 
[[Category:Reference]]
 
 
 
= Spectral Solution of Time Dependent Shallow Water Hydroelasticity =
 
\author{M\ls I\ls C\ls H\ls A\ls E\ls L\ns H.\ns M\ls E\ls Y\ls L\ls A\ls N<math>
 
^{1}</math>}
 
?? and in revised form ??
 
 
 
 
The spectral theory of a thin plate floating on shallow water is derived and
 
used to solve the time-dependent motion. This theory is based on an energy
 
inner product in which the evolution operator becomes unitary. Two solution
 
methods are presented. In the first, the solution is expanded in the
 
eigenfunctions of a self-adjoint operator, which are the incoming wave
 
solutions for a single frequency. In the second, the scattering theory of
 
Lax-Phillips is used. The Lax-Phillips scattering solution is suitable for
 
calculating only the free motion of the plate. However, it determines the
 
modes of vibration of the plate-water system. These modes, both oscillate
 
and decay, are found by a complex search algorithm based contour
 
integration. As well as an application to modelling floating runways, the
 
spectral-theory for a floating thin plate on shallow water is a solvable
 
model for more complicated hydroelastic systems.
 
 
 
==Introduction==
 
 
Hydroelasticity is the study of immersed or floating elastic bodies in a
 
fluid. It has a wide range of applications including very large floating
 
structures ([[Kashiwagihydro98]]), ships ([[Bishop]]), breakwaters (
 
[[Stoker]]) and sea ice ([[Squire_Review]]). One of the best studied
 
hydroelastic models is the linear floating thin plate ([[Newman_deform]],
 
[[ohmatsuVLFS]], [[Kagemoto97]], and [[Kashiwagihydro98]]) because
 
it models many physical systems, such as a floating runway or an ice floe (
 
[[jgrfloecirc]]).
 
 
The time-dependence in linear hydroelastic problems is usually removed by
 
solving for a single frequency which we will refer to as the \emph{single
 
frequency solution}. The solution is normally found by expanding the elastic
 
body motion in the free modes of vibration and solving the fluid equations
 
using a Green function ([[Bishop]]). This is analogous to solving for a
 
rigid floating body using the six rigid modes ([[Sarp_Isa]]). While
 
alternative methods have been developed ([[Kashiwagibspline]], \cite
 
{ohmatsuVLFS}, and [[Kagemoto97]]), these are based on exploiting some
 
property such as a simple geometry or high relative stiffness.
 
 
In contrast to the single frequency solutions, solving time-dependent linear
 
hydroelastic systems remains a major challenge. [[Kashiwagitime]] and
 
[[Endotime]] have developed a time-stepping procedure; however, this
 
method results in error growth in time. Since the problem is linear it is
 
solvable by a spectral method which eliminates the long-time growth of
 
errors. Furthermore, such a method provides information about the behaviour
 
of the solution, such as the decay constant of the motion. However, the
 
spectral theory for linear hydroelasticity has not been developed. For this
 
reason, spectral type solutions such as [[Ohmatsutimedep]], based on a
 
Fourier expansion of the single frequency solution, have only solved the
 
problem in restricted circumstances.
 
 
A floating thin plate on shallow water is considered in this paper. This
 
problem has been chosen for the following reasons: while the single
 
frequency solution is straightforward ([[Stoker]]), the time-dependent
 
solution has never been calculated; recently [[OhkusuNamba]], \cite
 
{OhkusuISOPE}, and [[Ertekinshallow1999]] used it to model a floating
 
runway; the spectral-theory developed here is a solvable model for more
 
complex hydroelastic systems.
 
 
The spectral theory for a thin plate on shallow water is based on an inner
 
product which gives the energy of the plate-water system. With respect to
 
this inner product the evolution operator becomes unitary. Two different
 
solution methods are derived from this spectral theory. The first method is
 
based on an expansion of the solution in eigenfunctions of a self-adjoint
 
operator. These are the single frequency solutions. The second solution
 
method is based on the scattering theory of Lax-Phillips ([[laxphilips]]
 
). It provides the solution in terms of a countable number of modes which
 
have both an oscillation and a decay. These modes are important as they can
 
be used to characterise the response of the system. With the exception of
 
[[Hazard]], they have not been investigated for hydroelastic systems.
 
 
==Formulation: A Thin Plate on Shallow Water==
 
 
Figure \ref{shallow} shows a schematic diagram of the problem. The plate is
 
infinite in the <math>y</math> direction, so that only the <math>x</math> and <math>z</math> directions are
 
considered. The <math>x</math> direction is horizontal, the positive <math>z</math> axis points
 
vertically up, and the plate covers the region <math>-b\leqslant x\leqslant b.</math>
 
The water is of uniform depth <math>h</math> which is small<math>\ </math>enough that the water
 
may be approximated as shallow ([[Stoker]]). The amplitudes are assumed
 
small enough that the linear theory is appropriate, and the plate is
 
sufficiently thin that the shallow draft approximation may be made (\cite
 
{OhkusuISOPE}). The solution could be extended to waves incident at an angle
 
on a infinite two dimensional plate, as described in [[OhkusuISOPE]], but
 
to keep the treatment straightforward this is not done here.
 
 
The mathematical description of the problem follows from [[Stoker]]. The
 
kinematic condition is
 
<center><math>
 
\partial _{t}\zeta =-h\partial _{x}^{2}\phi ,  (1)
 
</math></center>
 
where <math>\phi </math> is the velocity potential of the water (averaged over the
 
depth) and <math>\zeta </math> is the displacement of the water surface or the plate
 
(from the shallow draft approximation). The equation derived by equating the
 
pressure at the free surface is
 
<center><math>
 
-\rho g\zeta -\rho \partial _{t}\phi =\left\{
 
\begin{matrix}{c}
 
0,\;\;x\notin (-b,b), \\
 
D\partial _{x}^{4}\zeta +\rho ^{\prime }d\partial _{t}^{2}\zeta ,\;\;x\in
 
(-b,b),
 
\end{matrix}
 
\right.  (2)
 
</math></center>
 
where <math>D</math> is the bending rigidity of the plate per unit length, <math>\rho </math> is
 
the density of water, <math>\rho ^{\prime }</math> is the average density of the plate,
 
<math>g</math> is the acceleration due to gravity, and <math>d</math> is the thickness of the plate
 
<math>.</math> At the ends of the plate the free edge boundary conditions
 
<center><math>
 
\lim_{x\downarrow -b}\partial _{x}^{2}\zeta =\lim_{x\uparrow b}\partial
 
_{x}^{2}\zeta =\lim_{x\downarrow -b}\partial _{x}^{3}\zeta =\lim_{x\uparrow
 
b}\partial _{x}^{3}\zeta =0  (3)
 
</math></center>
 
are applied since these are common in offshore engineering applications (
 
[[OhkusuISOPE]]). However the theory which will be developed applies
 
equally to any of the energy-conserving edge conditions such as clamped or
 
pinned and there is no need for the boundary conditions to be symmetric.
 
Equation (3) gives the following implied boundary
 
conditions for <math>\phi </math>
 
<center><math>
 
\lim_{x\downarrow -b}\partial _{x}^{4}\phi =\lim_{x\uparrow b}\partial
 
_{x}^{4}\phi =\lim_{x\downarrow -b}\partial _{x}^{5}\phi =\lim_{x\uparrow
 
b}\partial _{x}^{5}\phi =0  (4)
 
</math></center>
 
which will be used subsequently.
 
 
Non-dimensional variables are now introduced. The space variables are
 
non-dimensionalised using the water depth <math>h,</math> and the time variables are
 
non-dimensionalised using <math>\sqrt{h/g}</math>. The non-dimensional variables are
 
<center><math>
 
\bar{x}=\frac{x}{h},\;\bar{t}=t\sqrt{\frac{g}{h}},\;\bar{\zeta}=\frac{\zeta
 
}{h},\;\mathrm{and}\;\bar{\phi}=\frac{\phi }{h^{2}\sqrt{g/h}}.
 
</math></center>
 
In these new variables, (1) and (2) become
 
<center><math>
 
\partial _{\bar{t}}\bar{\zeta}=-\partial _{\bar{x}}^{2}\bar{\phi}
 
(5)
 
</math></center>
 
and
 
<center><math>
 
-\bar{\zeta}-\partial _{\bar{t}}\bar{\phi}=\left\{
 
\begin{matrix}{c}
 
0,\;\;\bar{x}\notin (-\bar{b},\bar{b}), \\
 
\beta \partial _{\bar{x}}^{4}\bar{\zeta}+\gamma \partial _{\bar{t}}^{2}\bar{
 
\zeta},\;\;\bar{x}\in (-\bar{b},\bar{b}),
 
\end{matrix}
 
\right.  (6)
 
</math></center>
 
where <math>\beta </math> and <math>\gamma </math> are
 
<center><math>
 
\beta =\frac{D}{\rho gh^{4}}\;\;\mathrm{and\ \ }\gamma =\frac{\rho ^{\prime
 
}d}{\rho h}.
 
</math></center>
 
For clarity the overbar is dropped from now on.
 
 
The main change in extending the formulation to water of finite depth is
 
that the velocity potential will be governed by Laplace's equation. This
 
makes the solution of the problem much more computationally demanding since
 
Laplace's equation must be solved by a numerical method, for example the
 
boundary element method. Furthermore, the extension of the spectral theory,
 
which will be developed here for shallow water, to water of finite depth is
 
non-trivial and remains a subject for further work.
 
 
===Neglecting the inertia term.===
 
 
It can be assumed that <math>\left| \gamma \partial _{t}^{2}\zeta \right| \ll
 
\left| \zeta \right| </math> for the following reasons ([[OhkusuISOPE]]). If we
 
consider a mode of the displacement <math>\zeta =ae^{i\omega t}</math> (where <math>a</math> is
 
the amplitude) then <math>\partial _{t}^{2}\zeta =-\omega ^{2}ae^{i\omega t}.</math>
 
For each frequency, <math>\omega ,</math> there is a corresponding wavelength <math>\lambda
 
=2\pi /\omega .</math> In the non-dimensional variables the wave speed and water
 
depth are both unity. Since the water is shallow the wavelength <math>\lambda \gg
 
1<math> and thus </math>\omega \ll 1.</math> It follows that any shallow water mode must
 
satisfy <math>\left| \partial _{t}^{2}\zeta \right| \ll \left| \zeta \right| .</math>
 
Also, <math>\gamma \ll 1</math> since <math>\rho ^{\prime }<\rho </math> (otherwise the plate
 
would sink) and <math>d\ll h</math> (otherwise the submergence of the plate would not
 
be negligible). Therefore,
 
<center><math>
 
\left| \gamma \partial _{t}^{2}\zeta \right| \ll \left| \zeta \right| ,
 
</math></center>
 
and we assume in what follows that the inertia, <math>\gamma \partial
 
_{t}^{2}\zeta ,</math> is zero.
 
 
It should be noted that the inclusion of the inertia term in the spectral
 
theory which will be developed is difficult because it introduces a time
 
dependence in the energy inner product.
 
 
==The Energy Inner Product==
 
 
While equations (5) and (6) are not
 
complicated they cannot be solved in a simple manner. It is not possible to
 
Fourier transform in space because of the spatial discontinuity of the
 
differential equations. The Weiner-Hopf technique cannot be used because the
 
discontinuities divide the space into three regions. A Laplace
 
transformation in time can be applied but this leads to non-trivial
 
equations involving a spatially discontinuous differential equation subject
 
to arbitrary initial conditions. However, straightforward solutions can be
 
derived using spectral theory.
 
 
The spectral-theory solution of equations (5) and (\ref
 
{displacement2}) is based on the spectral theory for a unitary operator
 
(essentially, an operator is unitary if the adjoint is also the inverse). We
 
therefore require an inner product in which the evolution operator is
 
unitary. This inner product, since the system is conservative, is derived
 
from the energy. The potential and displacement both contribute to this
 
energy and we combine them in a two component vector, <math>U\left( x,t\right) </math>,
 
given by
 
<center><math>
 
U\left( x,t\right) =\left(
 
\begin{matrix}{c}
 
\phi (x,t) \\
 
i\zeta (x,t)
 
\end{matrix}
 
\right) .  (7)
 
</math></center>
 
The energy consists of the kinetic energy of the water (<math>\propto \left| \phi
 
_{t}^{2}\right| <math>), the potential energy of the water (</math>\propto \left| \phi
 
^{2}\right| </math>), and the energy of the plate. The energy inner product for
 
the two vectors
 
<center><math>
 
U_{1}=\left(
 
\begin{matrix}{c}
 
\phi _{1} \\
 
i\zeta _{1}
 
\end{matrix}
 
\right) \;\;\mathrm{and\ \ }U_{2}=\left(
 
\begin{matrix}{c}
 
\phi _{2} \\
 
i\zeta _{2}
 
\end{matrix}
 
\right)
 
</math></center>
 
is given by
 
<center><math>
 
\left\langle U_{1},U_{2}\right\rangle _{\mathcal{H}}=\left\langle \partial
 
_{x}\phi _{1},\partial _{x}\phi _{2}\right\rangle +\left\langle \left(
 
1+\beta \left( H\left( x-b\right) -H\left( x+b\right) \right) \partial
 
_{x}^{4}\right) i\zeta _{1},i\zeta _{2}\right\rangle ,
 
(8)
 
</math></center>
 
where <math>H</math> is the Heaviside function. The subscript <math>\mathcal{H}</math> is used to
 
denote the special inner product and the angle brackets without the <math>
 
\mathcal{H}</math> denote the standard inner product, i.e.
 
<center><math>
 
\left\langle f\left( x\right) ,g\left( x\right) \right\rangle =\int_{-\infty
 
}^{\infty }f\left( x\right) g^{\ast }\left( x\right) dx.
 
</math></center>
 
We now write (5) and (6) as
 
<center><math>\begin{matrix}
 
\frac{1}{i}\partial _{t}U &=&\mathcal{P}U  (9) \\
 
U\left( x,t\right) _{t=0} &=&U_{0}\left( x\right) =\left(
 
\begin{matrix}{c}
 
\phi _{0}(x) \\
 
i\zeta _{0}(x)
 
\end{matrix}
 
\right)  \nonumber
 
\end{matrix}</math></center>
 
where the operator <math>\mathcal{P}</math> is
 
<center><math>
 
\mathcal{P=}\left(
 
\begin{matrix}{cc}
 
0 & 1+\beta \left( H\left( x-b\right) -H\left( x+b\right) \right) \partial
 
_{x}^{4} \\
 
-\partial _{x}^{2} & 0
 
\end{matrix}
 
\right) .
 
</math></center>
 
<math>\mathcal{P}</math> is self-adjoint with respect to the inner product (\ref
 
{energyinnerprod}) since <math>\mathcal{P}</math> satisfies
 
<center><math>
 
\left\langle \mathcal{P}U_{1},U_{2}\right\rangle _{\mathcal{H}}=\left\langle
 
U_{1},\mathcal{P}U_{2}\right\rangle _{\mathcal{H}}
 
</math></center>
 
from integration by parts and the boundary conditions at the end of the
 
plate (3). We can express the solution to (\ref
 
{selfadjoint2}) as
 
<center><math>
 
U\left( x,t\right) =e^{i\mathcal{P}t}U_{0}  (10)
 
</math></center>
 
where <math>e^{i\mathcal{P}t}</math> is a unitary evolution operator.
 
 
==The self-adjoint solution method(11)==
 
 
In this section, a solution for the time dependent motion of the plate-water
 
system is developed using the theory of self-adjoint operators. To evaluate
 
equation (10) we require a method to calculate the evolution
 
operator <math>e^{i\mathcal{P}t}</math>. This can be accomplished by using the
 
eigenfunctions of the operator <math>\mathcal{P},</math> which are the single frequency
 
solutions.
 
 
===Finding the eigenfunctions(12)===
 
 
Since <math>\mathcal{P}</math> is self-adjoint, the eigenvalues, <math>\lambda ,</math> must be
 
real and therefore<math>\ </math>the eigenfunctions of <math>\mathcal{P}</math> are oscillatory
 
exponentials outside the region of water covered by the plate. Furthermore,
 
since the plate is finite, the spectrum (set of eigenvalues) is the entire
 
real numbers. As is expected for two-component systems, there are two
 
eigenfunctions associated with each eigenvalue <math>\lambda </math>. We choose
 
incoming waves from the left (<math>\Phi ^{>})</math> and the right (<math>\Phi ^{<})</math> of
 
unit amplitude as a basis for the eigenspace since they are the standard
 
single frequency solutions. They have the following asymptotics,
 
<center><math>
 
\lim_{x\rightarrow -\infty }\Phi ^{>}=\left(
 
\begin{matrix}{c}
 
e^{i\lambda x} \\
 
\lambda e^{i\lambda x}
 
\end{matrix}
 
\right) +S_{12}\left(
 
\begin{matrix}{c}
 
e^{-i\lambda x} \\
 
\lambda e^{-i\lambda x}
 
\end{matrix}
 
\right) \;\;\mathrm{and\ \ }\lim_{x\rightarrow \infty }\Phi
 
^{>}=S_{11}\left(
 
\begin{matrix}{c}
 
e^{i\lambda x} \\
 
\lambda e^{i\lambda x}
 
\end{matrix}
 
\right)
 
</math></center>
 
and
 
<center><math>
 
\lim_{t\rightarrow -\infty }\Phi ^{<}=S_{22}\left(
 
\begin{matrix}{c}
 
e^{-i\lambda x} \\
 
\lambda e^{-i\lambda x}
 
\end{matrix}
 
\right) \;\;\mathrm{and\ \ }\lim_{x\rightarrow \infty }\Phi ^{>}=\left(
 
\begin{matrix}{c}
 
e^{-i\lambda x} \\
 
\lambda e^{-i\lambda x}
 
\end{matrix}
 
\right) +S_{21}\left(
 
\begin{matrix}{c}
 
e^{i\lambda x} \\
 
\lambda e^{i\lambda x}
 
\end{matrix}
 
\right) ,
 
</math></center>
 
where <math>S_{11}</math>, <math>S_{12,}</math> <math>S_{21},</math> and <math>S_{22}</math> are the reflection and
 
transmission coefficients (which must be determined). These eigenfunctions,
 
which are analogous to the Jost solutions of Schrodinger's equation (\cite
 
{Chadan89}), will be used to calculated the time-dependent solution.
 
 
We find the eigenfunction <math>\Phi ^{>}(\lambda ,x)</math> by solving (\ref
 
{kinematic2}) and (6) in each region. The two components, <math>
 
\phi ^{>}\left( \lambda ,x\right) <math> and </math>i\zeta ^{>}\left( \lambda ,x\right)
 
,</math> are given by
 
<center><math>
 
\phi ^{>}\left( \lambda ,x\right) =\left\{
 
\begin{matrix}{c}
 
e^{-i\lambda x}+S_{11}\left( \lambda \right) e^{i\lambda x},\;\;x<-b, \\
 
\sum\limits_{j=1}^{6}\alpha _{j}e^{\mu _{j}\left( \lambda \right)
 
x},\;\;-b<x<b, \\
 
S_{12}\left( \lambda \right) e^{-i\lambda x},\;\;x>b,
 
\end{matrix}
 
\right.  (13)
 
</math></center>
 
and
 
<center><math>
 
i\zeta ^{>}\left( \lambda ,x\right) =\left\{
 
\begin{matrix}{c}
 
\lambda e^{-i\lambda x}+\lambda S_{11}\left( \lambda \right) e^{i\lambda
 
x},\;\;x<-b, \\
 
\frac{-1}{\lambda }\sum\limits_{j=1}^{6}\mu _{j}\left( \lambda \right)
 
^{2}\alpha _{j}e^{\mu _{j}\left( \lambda \right) x},\;\;-b<x<b, \\
 
\lambda S_{12}\left( \lambda \right) e^{-i\lambda x},\;\;x>b,
 
\end{matrix}
 
\right.
 
</math></center>
 
where the coefficients <math>\mu _{j}\left( \lambda \right) </math> are the six roots
 
of the equation
 
<center><math>
 
\beta \mu ^{6}+\mu ^{2}+\lambda ^{2}=0.  (14)
 
</math></center>
 
The values of <math>S_{11}\left( \lambda \right) ,</math> <math>S_{12}\left( \lambda \right)
 
,<math> and </math>\alpha _{j}</math> are found from the boundary conditions (4)
 
and the continuity of <math>\phi </math> and <math>\partial _{x}\phi </math> at <math>x=\pm b.</math>
 
Therefore, to find the eigenfunction <math>\Phi ^{>}\left( \lambda ,x\right) ,</math>
 
we solve the 8 by 8 linear system
 
<center><math>
 
\mathbf{M}\vec{a}\mathbf{=}\vec{b},  (15)
 
</math></center>
 
where <math>\mathbf{M}</math> is the matrix
 
<center><math>
 
\mathbf{M}=\left(
 
\begin{matrix}{cccccccc}
 
\mu _{1}^{4}e^{-\mu _{1}b} & \mu _{2}^{4}e^{-\mu _{2}b} & \mu
 
_{3}^{4}e^{-\mu _{3}b} & \mu _{4}^{4}e^{-\mu _{4}b} & \mu _{5}^{4}e^{-\mu
 
_{5}b} & \mu _{6}^{4}e^{-\mu _{6}b} & 0 & 0 \\
 
\mu _{1}^{5}e^{-\mu _{1}b} & \mu _{2}^{5}e^{-\mu _{2}b} & \mu
 
_{3}^{5}e^{-\mu _{3}b} & \mu _{4}^{5}e^{-\mu _{4}b} & \mu _{5}^{5}e^{-\mu
 
_{5}b} & \mu _{6}^{5}e^{-\mu _{6}b} & 0 & 0 \\
 
\mu _{1}^{4}e^{\mu _{1}b} & \mu _{2}^{4}e^{\mu _{2}b} & \mu _{3}^{4}e^{\mu
 
_{3}b} & \mu _{4}^{4}e^{\mu _{4}b} & \mu _{5}^{4}e^{\mu _{5}b} & \mu
 
_{6}^{4}e^{\mu _{6}b} & 0 & 0 \\
 
\mu _{1}^{5}e^{\mu _{1}b} & \mu _{2}^{5}e^{\mu _{2}b} & \mu _{3}^{5}e^{\mu
 
_{3}b} & \mu _{4}^{5}e^{\mu _{4}b} & \mu _{5}^{5}e^{\mu _{5}b} & \mu
 
_{6}^{5}e^{\mu _{6}b} & 0 & 0 \\
 
e^{-\mu _{1}b} & e^{-\mu _{2}b} & e^{-\mu _{3}b} & e^{-\mu _{4}b} & e^{-\mu
 
_{5}b} & e^{-\mu _{6}b} & -e^{-i\lambda b} & 0 \\
 
\mu _{1}e^{-\mu _{1}b} & \mu _{2}e^{-\mu _{2}b} & \mu _{3}e^{-\mu _{3}b} &
 
\mu _{4}e^{-\mu _{4}b} & \mu _{5}e^{-\mu _{5}b} & \mu _{6}e^{-\mu _{6}b} &
 
-i\lambda e^{-i\lambda b} & 0 \\
 
e^{\mu _{1}b} & e^{\mu _{2}b} & e^{\mu _{3}b} & e^{\mu _{4}b} & e^{\mu _{5}b}
 
& e^{\mu _{6}b} & 0 & -e^{-i\lambda b} \\
 
\mu _{1}e^{\mu _{1}b} & \mu _{2}e^{\mu _{2}b} & \mu _{3}e^{\mu _{3}b} & \mu
 
_{4}e^{\mu _{4}b} & \mu _{5}e^{\mu _{5}b} & \mu _{6}e^{\mu _{6}b} & 0 &
 
i\lambda e^{-i\lambda b}
 
\end{matrix}
 
\right)
 
</math></center>
 
and <math>\vec{a}</math> and <math>\vec{b}</math> are given by
 
<center><math>
 
\vec{a}=\left(
 
\begin{matrix}{c}
 
\alpha _{1} \\
 
\alpha _{2} \\
 
\alpha _{3} \\
 
\alpha _{4} \\
 
\alpha _{5} \\
 
\alpha _{6} \\
 
S_{11} \\
 
S_{12}
 
\end{matrix}
 
\right) ,\;\;\;\;\;\vec{b}=\left(
 
\begin{matrix}{c}
 
0 \\
 
0 \\
 
0 \\
 
0 \\
 
e^{i\lambda b} \\
 
-i\lambda e^{i\lambda b} \\
 
0 \\
 
0
 
\end{matrix}
 
\right) .
 
</math></center>
 
Note that the coefficients <math>S_{11}</math> and <math>S_{12}</math> are contained in <math>\vec{a}.</math>
 
 
The eigenfunctions for the wave propagating from the right <math>\Phi ^{<}\left(
 
\lambda ,x\right) <math> are found similarly. Since </math>S_{11}</math> represents the
 
amplitude of the reflected wave and <math>S_{12}</math> represents the amplitude of the
 
transmitted wave, conservation of energy requires that <math>\left| S_{11}\right|
 
^{2}+\left| S_{12}\right| ^{2}=1.</math> Similarly, since the boundary conditions
 
are symmetric <math>S_{22}\left( \lambda \right) =S_{11}\left( \lambda \right) </math>
 
and <math>S_{12}\left( \lambda \right) =S_{21}\left( \lambda \right) .</math>
 
 
===Solution with the eigenfunctions===
 
 
Equation (10) can be solved by a generalised Fourier transform
 
based on the eigenfunctions of the operator <math>\mathcal{P}</math>. The
 
eigenfunctions are orthogonal since <math>\mathcal{P}</math> is self-adjoint, but they
 
must be normalised. This is accomplished by using the following identity
 
<center><math>
 
\int_{0}^{\infty }e^{i\left( \lambda _{1}-\lambda _{2}\right) t}dt=\pi
 
\delta \left( \lambda _{1}-\lambda _{2}\right)
 
</math></center>
 
where <math>\delta </math> is the Dirac delta function. Therefore
 
<center><math>\begin{matrix}
 
\left\langle \Phi ^{>}\left( x,\lambda _{1}\right) ,\Phi ^{>}\left(
 
x,\lambda _{2}\right) \right\rangle _{\mathcal{H}} &=&\pi \delta \left(
 
\lambda _{1}-\lambda _{2}\right) \lambda _{1}^{2}\left( 1+\left|
 
S_{11}\right| ^{2}+\left| S_{12}\right| ^{2}\right) \\
 
&&+\pi \delta \left( \lambda _{1}-\lambda _{2}\right) \lambda _{1}\lambda
 
_{2}\left( 1+\left| S_{11}\right| ^{2}+\left| S_{12}\right| ^{2}\right) \\
 
&=&4\pi \delta \left( \lambda _{1}-\lambda _{2}\right) \lambda _{1}^{2},
 
\nonumber
 
\end{matrix}</math></center>
 
using the condition <math>\left| S_{11}\right| ^{2}+\left| S_{12}\right| ^{2}=1.</math>
 
Similarly
 
<center><math>
 
\left\langle \Phi ^{<}\left( x,\lambda _{1}\right) ,\Phi ^{<}\left(
 
x,\lambda _{2}\right) \right\rangle _{\mathcal{H}}=4\pi \delta \left(
 
\lambda _{1}-\lambda _{2}\right) \lambda _{1}^{2}
 
</math></center>
 
and
 
<center><math>\begin{matrix}
 
\left\langle \Phi ^{>}\left( x,\lambda _{1}\right) ,\Phi ^{<}\left(
 
x,\lambda _{2}\right) \right\rangle _{\mathcal{H}} &=&2\pi \delta \left(
 
\lambda _{1}-\lambda _{2}\right) \lambda _{1}^{2}\left( S_{11}S_{21}^{\ast
 
}+S_{12}S_{22}^{\ast }\right) \\
 
&=&0.  \nonumber
 
\end{matrix}</math></center>
 
 
The generalised Fourier transform which solves the evolution equation (\ref
 
{unitary}) is
 
<center><math>\begin{matrix}
 
U\left( x,t\right) &=&\int_{-\infty }^{\infty }\left\langle U_{0}\left(
 
x\right) ,\frac{\Phi ^{>}\left( x,\lambda \right) }{4\pi \lambda ^{2}}
 
\right\rangle _{\mathcal{H}}\Phi ^{>}\left( x,\lambda \right) e^{i\lambda
 
t}d\lambda  (16) \\
 
&&+\int_{-\infty }^{\infty }\left\langle U_{0}\left( x\right) ,\frac{\Phi
 
^{<}\left( x,\lambda \right) }{4\pi \lambda ^{2}}\right\rangle _{\mathcal{H}
 
}\Phi ^{<}\left( x,\lambda \right) e^{i\lambda t}d\lambda .  \nonumber
 
\end{matrix}</math></center>
 
Equation (16) is the cornerstone of the approach. The
 
integral in equation (16) can be calculated by the fast
 
Fourier transform while the inner product can calculated by the fast Fourier
 
transform if the initial condition <math>U_{0}</math> is zero underneath the plate (<math>
 
-b<x<b).</math>
 
 
===Numerical Calculations===
 
 
The intention of this paper is to develop the solution methods rather than
 
describe the physics of the motion and therefore only a few solutions are
 
presented. From [[OhkusuISOPE]] we assume the plate stiffness is <math>\beta
 
=2\times 10^{4}<math> and the plate length is </math>b=50</math> throughout. These values are
 
typical for a floating runway. Figures \ref{incomingspecplot1} and \ref
 
{incomingspecplot2} show the displacement and potential, respectively, for a
 
pulse travelling to the right at the times <math>t=0,</math> 30, 60, 90, 120, 150, 180,
 
210, and 240. The incoming wave pulse was chosen to be a Gaussian in
 
potential centered at <math>x=-125</math> and sufficiently sharp to be negligible under
 
the plate,
 
<center><math>
 
U_{0}\left( x\right) =\left(
 
\begin{matrix}{c}
 
\phi \left( x\right) \\
 
i\phi ^{\prime }\left( x\right)
 
\end{matrix}
 
\right)
 
</math></center>
 
where
 
<center><math>
 
\phi \left( x\right) =\left\{
 
\begin{matrix}{c}
 
e^{-\tfrac{(x+125)^{2}}{350}},\;\;x<-50, \\
 
0,\;\;x>-50.
 
\end{matrix}
 
\right.
 
</math></center>
 
At <math>t=0</math> the plate is initially at rest and the wave is to the left of the
 
plate propagating towards it. From <math>t=30</math> the wave has reached the plate and
 
the plate begins to undergo a complex bending motion in response to the
 
incoming wave. The response of the plate in turn induces waves in the
 
surrounding water which propagate away from the plate to the left and right.
 
The final picture, <math>t=240</math>, shows the plate at rest with waves now
 
propagating away from it. The majority of the wave energy has passed under
 
the plate and continues to propagate to the right. However, the shape of the
 
outgoing wave profile is markedly different from the incoming wave profile.
 
Also, there is a significant reflected wave propagating away from the plate
 
to the left.
 
 
Figures \ref{spectral1} and \ref{spectral2} show the evolution of the plate
 
from an initial displacement in the absence of wave forcing for the times <math>
 
t=0,</math> 20, 40, 60, 80, 100, 120, 140, and 160. Only the plate displacement is
 
initially non-zero so that
 
<center><math>
 
U_{0}\left( x\right) =\left(
 
\begin{matrix}{c}
 
0 \\
 
i\zeta \left( x\right)
 
\end{matrix}
 
\right) .
 
</math></center>
 
Figure \ref{spectral1} shows the motion for the symmetric initial plate
 
displacement
 
<center><math>
 
\zeta \left( x\right) =e^{-\tfrac{x^{2}}{350}}.
 
</math></center>
 
As the plate evolves the plate vibrates, straightens, and the amplitude
 
decays. The decay is due to the radiation of energy by the waves generated
 
in the surrounding water. A complex wave train is produced by the plate
 
motion and can be seen propagating away from the plate. Figure \ref
 
{spectral2} shows the motion for the non-symmetric initial plate
 
displacement
 
<center><math>
 
\zeta \left( x\right) =e^{-\tfrac{\left( x-50\right) ^{2}}{350}}.
 
</math></center>
 
Again as the plate evolves it straightens, vibrates, and decays and induces
 
waves in the surrounding water.
 
 
==The Lax-Phillips Scattering Solution Method.==
 
 
In this section, a solution to the time-dependent motion of the plate-water
 
system is developed using the Lax-Phillips scattering theory (\cite
 
{laxphilips}). This solution method will only solve for an initial condition
 
which is zero outside the plate, i.e. <math>U_{0}\left( x\right) =0</math> if <math>\left|
 
x\right| >b</math>. However, it calculates the solution by an expansion in a
 
countable number of modes.
 
 
===Lax-Phillips Scattering(17)===
 
 
The Lax-Phillips scattering theory will be briefly outlined here for our
 
specific problem. The Hilbert space <math>\mathcal{H}</math>\ is decomposed into three
 
subspaces. The incoming space, denoted by <math>D_{-},</math> consists of all waves
 
travelling towards the plate, either from the left or the right, as
 
appropriate. The outgoing subspace, denoted by <math>D_{+},</math> consists of all
 
waves travelling away from the plate, again either to the left or right, as
 
appropriate. What remains is the scattering space, denoted by <math>K,\ </math>
 
consisting of the potential and displacement under the plate.
 
 
To apply the Lax-Phillips scattering the following conditions are required: <math>
 
D_{-}<math> and </math>D_{+}</math> must be orthogonal; the incoming subspace must span the
 
entire space under temporal evolution. For our system, the first condition
 
follows from the inner product and the second condition follows from the
 
simple structure of the eigenfunctions of the operator <math>\mathcal{P}.</math> From
 
the Lax-Phillips scattering theory, since these conditions hold, the
 
equation of motion for the plate in the absence of incoming waves can be
 
written
 
<center><math>
 
\frac{1}{i}\partial _{t}U=\mathcal{B}U,  (18)
 
</math></center>
 
where <math>\mathcal{B}</math> is a non-self-adjoint operator. <math>\mathcal{B}</math> is related
 
to <math>\mathcal{P}</math> by
 
<center><math>
 
e^{i\mathcal{B}t}=P_{K}\left. e^{i\mathcal{P}t}\right| _{K}
 
</math></center>
 
where <math>P_{K}</math> is the projection onto the subspace <math>K</math> and <math>\left. {}\right|
 
_{K}<math> denotes a restriction to </math>K.<math> Therefore </math>e^{i\mathcal{B}t}</math> is the
 
restricted to <math>K</math> of the evolution of an initial condition which is zero
 
outside <math>K.</math> It should be noted that the equality in equation (\ref
 
{nonselffirst}) is in general only true asymptotically. However the
 
numerical results show for our case we have equality for all times.
 
 
From the Lax-Phillips scattering theory, equation (18) can
 
be solved by finding the eigenvalues (sometimes referred to as scattering
 
frequencies or resonances) and eigenfunctions of <math>\mathcal{B}</math>. The
 
eigenvalues of <math>\mathcal{B}</math> occur at the singularities of the analytic
 
extension to <math>\mathbb{C}</math> of the scattering matrix, <math>\mathbf{S}(\lambda ).</math>
 
This is given by
 
<center><math>
 
\mathbf{S}(\lambda )=\left(
 
\begin{matrix}{cc}
 
S_{11}\left( \lambda \right) & S_{12}\left( \lambda \right) \\
 
S_{21}\left( \lambda \right) & S_{22}\left( \lambda \right)
 
\end{matrix}
 
\right)  (19)
 
</math></center>
 
where <math>S_{11}\left( \lambda \right) </math>, <math>S_{12}\left( \lambda \right) ,</math> <math>
 
S_{21}\left( \lambda \right) ,<math> and </math>S_{22}\left( \lambda \right) </math> are the
 
scattered wave coefficients found from the single frequency solutions in
 
section 12. As a consequence of the Lax-Phillips scattering
 
theory the scattering matrix is unitary for real <math>\lambda </math> and the
 
singularities must all lie in the upper complex plane (<math>{Im}\left(
 
\lambda \right) >0).</math> Once the singularities have been found, the
 
eigenfunctions can be calculated. They are not orthogonal, since <math>\mathcal{B}
 
</math> is a non-self-adjoint operator, but a biorthogonal system can be formed
 
using the eigenfunctions of the adjoint operator, <math>\mathcal{B}^{\ast }.</math>
 
 
The eigenfunctions of <math>\mathcal{B}</math> are the modes of vibration for the
 
plate-water system. These modes have a decay as well as an oscillation due
 
to the radiation of energy into the surrounding water. The frequency of the
 
oscillation is determined by the real part of the eigenvalue and the rate of
 
decay is determined by the imaginary part of the eigenvalue.
 
 
While the eigenvalues of <math>\mathcal{B}</math> occur precisely at the singularities
 
of the solution found by a Laplace transformation in time the Lax-Phillips
 
scattering theory solution has three major advantages over the Laplace
 
transform solution: the eigenvalues (singularities) can be found using the
 
scattering matrix; the difficult equations in the Laplace space involving
 
the initial condition do not need to be solved; the contribution of the
 
singularity (the residue) can be found directly from the inner product of
 
the initial condition with the corresponding eigenfunction of the adjoint
 
operator, <math>\mathcal{B}^{\ast }.</math>
 
 
===Finding the Singularities of the Scattering Matrix===
 
 
While the analytic extension of the scattering matrix is straightforward,
 
the linear system (15) is solved for complex <math>\lambda ,</math>
 
finding the singularities of the scattering matrix is non-trivial<math>.</math> The
 
difficulty lies in the fact that we must search the complex plane for the
 
singularities with no ''a priori''  knowledge about their location. We use
 
a complex search algorithm based contour integration. The determinant of the
 
scattering matrix is integrated around the contour of a region of the
 
complex plane. If the value of this integral is zero, then the region is
 
assumed to contain no singularities and the search is terminated (the
 
possibility that the contribution of two or more singularities might cancel
 
can be treated by considering further integrals, such as the variation of
 
the argument around the contour). If the value of the integral is not zero,
 
then the region must contain singularities and it is then divided into
 
subregions and the search is repeated. Once the singularities have been
 
located sufficiently well they are used as seeds for Newton's method and
 
found to high accuracy.
 
 
If the eigenvalues have to be found for different parameter values then a
 
homotopy, or continuation, method can be used, which avoids the slow complex
 
search method. This method uses the known locations of the eigenvalues for
 
one parameter value to determine the eigenvalues for a new parameter value
 
by taking sufficiently small steps that Newton's method can be used with the
 
previous solution as a seed. Unfortunately, a homotopy method requires the
 
solution of the eigenvalues for at least one parameter value as an initial
 
seed and this must be accomplished by a complex search algorithm.
 
 
The position of the eigenvalues for <math>\beta =2\times 10^{4}</math> and <math>b=50</math> are
 
shown in Figure \ref{spectrum}. They are denoted by <math>\lambda _{n},</math> where <math>
 
n\in \mathbb{Z}<math>, and ordered by increasing real part, with </math>n=0</math>
 
corresponding the eigenvalue with smallest absolute real part. From the
 
picture and on physical grounds, it seems likely that there exist
 
asymptotics for the eigenvalues, however this theory is not developed here.
 
 
===Eigenfunctions===
 
 
The eigenfunctions of <math>\mathcal{B}</math> associated with the eigenvalue <math>\lambda
 
_{n}<math> are denoted by </math>\Phi ^{+}(\lambda _{n},x),<math> and those of </math>\mathcal{B}
 
^{\ast }<math> (the adjoint of </math>\mathcal{B})<math> associated with the eigenvalue </math>
 
\lambda _{n}^{\ast }<math> are denoted by </math>\hat{\Phi}^{+}\left( \lambda
 
_{n}^{\ast },x\right) </math>. That is,
 
<center><math>
 
\mathcal{B}\Phi ^{+}\left( \lambda _{n},x\right) =\lambda _{n}\Phi
 
^{+}\left( \lambda _{n},x\right)
 
</math></center>
 
and
 
<center><math>
 
\mathcal{B}^{\ast }\hat{\Phi}^{+}\left( \lambda _{n}^{\ast },x\right)
 
=\lambda _{n}^{\ast }\hat{\Phi}^{+}\left( \lambda _{n}^{\ast },x\right) .
 
</math></center>
 
The eigenfunction <math>\Phi ^{+}\left( \lambda _{n},x\right) </math> can be written
 
<center><math>
 
\Phi ^{+}\left( \lambda _{n},x\right) =\left(
 
\begin{matrix}{c}
 
\phi ^{+}\left( \lambda _{n},x\right) \\
 
i\zeta ^{+}\left( \lambda _{n},x\right)
 
\end{matrix}
 
\right) =\left(
 
\begin{matrix}{c}
 
\sum\limits_{j=1}^{6}\alpha _{j}e^{\mu _{j}\left( \lambda _{n}\right) x} \\
 
\sum\limits_{j=1}^{6}-\frac{\alpha _{j}\mu _{j}\left( \lambda _{n}\right)
 
^{2}}{\lambda _{n}}e^{\mu _{j}\left( \lambda _{n}\right) x}
 
\end{matrix}
 
\right)
 
</math></center>
 
where <math>\mu _{j}\left( \lambda \right) </math> are the six roots of equation (\ref
 
{cubicinlambda}) and the coefficients <math>\alpha _{j}</math> are found from the
 
boundary conditions at the end of the plate (4) and the
 
following condition. Since the scattering matrix is singular, there are no
 
incoming wave from either direction. We use the condition that there is no
 
incoming wave at <math>x=-b</math> and that the outgoing wave is of unit amplitude,
 
i.e.
 
<center><math>
 
\phi ^{+}\left( \lambda _{n},-b\right) =e^{i\lambda _{n}b},\;=\ = \left.
 
\partial _{x}\phi ^{+}\left( \lambda _{n},x\right) \right| _{x=-b}=i\lambda
 
_{n}e^{i\lambda _{n}b}.
 
</math></center>
 
We do not use the condition that there is no outgoing wave at <math>x=b</math> because
 
the system will become over determined. Therefore, the coefficients <math>\alpha
 
_{j}</math> satisfy the linear equation
 
<center><math>
 
\mathbf{M}\vec{a}\mathbf{=}\vec{b},
 
</math></center>
 
where
 
<center><math>
 
\mathbf{M}=\left(
 
\begin{matrix}{cccccc}
 
\mu _{1}^{4}e^{-\mu _{1}b} & \mu _{2}^{4}e^{-\mu _{2}b} & \mu
 
_{3}^{4}e^{-\mu _{3}b} & \mu _{4}^{4}e^{-\mu _{4}b} & \mu _{5}^{4}e^{-\mu
 
_{5}b} & \mu _{6}^{4}e^{-\mu _{6}b} \\
 
\mu _{1}^{5}e^{-\mu _{1}b} & \mu _{2}^{5}e^{-\mu _{2}b} & \mu
 
_{3}^{5}e^{-\mu _{3}b} & \mu _{4}^{5}e^{-\mu _{4}b} & \mu _{5}^{5}e^{-\mu
 
_{5}b} & \mu _{6}^{5}e^{-\mu _{6}b} \\
 
\mu _{1}^{4}e^{\mu _{1}b} & \mu _{2}^{4}e^{\mu _{2}b} & \mu _{3}^{4}e^{\mu
 
_{3}b} & \mu _{4}^{4}e^{\mu _{4}b} & \mu _{5}^{4}e^{\mu _{5}b} & \mu
 
_{6}^{4}e^{\mu _{6}b} \\
 
\mu _{1}^{5}e^{\mu _{1}b} & \mu _{2}^{5}e^{\mu _{2}b} & \mu _{3}^{5}e^{\mu
 
_{3}b} & \mu _{4}^{5}e^{\mu _{4}b} & \mu _{5}^{5}e^{\mu _{5}b} & \mu
 
_{6}^{5}e^{\mu _{6}b} \\
 
e^{-\mu _{1}b} & e^{-\mu _{2}b} & e^{-\mu _{3}b} & e^{-\mu _{4}b} & e^{-\mu
 
_{5}b} & e^{-\mu _{6}b} \\
 
\mu _{1}e^{-\mu _{1}b} & \mu _{2}e^{-\mu _{2}b} & \mu _{3}e^{-\mu _{3}b} &
 
\mu _{4}e^{-\mu _{4}b} & \mu _{5}e^{-\mu _{5}b} & \mu _{6}e^{-\mu _{6}b}
 
\end{matrix}
 
\right)
 
</math></center>
 
and <math>\vec{a}</math> and <math>\vec{b}</math> are given by
 
<center><math>
 
\vec{a}=\left(
 
\begin{matrix}{c}
 
\alpha _{1} \\
 
\alpha _{2} \\
 
\alpha _{3} \\
 
\alpha _{4} \\
 
\alpha _{5} \\
 
\alpha _{6}
 
\end{matrix}
 
\right) ,\;\;\;\;\;\vec{b}=\left(
 
\begin{matrix}{c}
 
0 \\
 
0 \\
 
0 \\
 
0 \\
 
e^{i\lambda b} \\
 
i\lambda e^{i\lambda b}
 
\end{matrix}
 
\right) .
 
</math></center>
 
The eigenfunctions for the adjoint operator are found similarly.
 
 
Figure \ref{eigfunctions} shows the real and imaginary parts of the
 
eigenfunctions of <math>\mathcal{B}</math> for <math>n=1,</math> <math>3</math>, 5, and <math>7,</math> again with <math>
 
\beta =2\times 10^{4}<math> and </math>b=50.</math> While the eigenfunctions do not have a
 
simple shape, increasing oscillation is apparent as <math>n</math> increases.
 
 
===Inner products===
 
 
A biorthogonal system with respect to the energy inner product (\ref
 
{energyinnerprod}) is formed by the eigenfunctions of <math>\mathcal{B}</math>, <math>\Phi
 
^{+}\left( \lambda _{n},x\right) ,<math> and the eigenfunctions of </math>\mathcal{B}
 
^{\ast },<math> </math>\hat{\Phi}^{+}\left( \lambda _{n},x\right) </math>. To normalise the
 
biorthogonal system, the inner product of <math>\Phi ^{+}\left( \lambda
 
_{n},x\right) <math> and </math>\hat{\Phi}^{+}\left( \lambda _{n},x\right) </math> has to be
 
determined. From the definition of the energy inner product (\ref
 
{energyinnerprod})
 
<center><math>\begin{matrix}
 
\left\langle \Phi \left( \lambda _{m},x\right) ,\hat{\Phi}\left( \lambda
 
_{n},x\right) \right\rangle _{\mathcal{H}} &=&\int_{-b}^{b}\partial _{x}\phi
 
^{+}\left( \lambda _{m},x\right) \left( \partial _{x}\hat{\phi}^{+}\left(
 
\lambda _{n}^{\ast },x\right) \right) ^{\ast }dx  (20) \\
 
&&+\int_{-b}^{b}(1+P)i\zeta ^{+}\left( \lambda _{m}^{\ast },x\right) \left( i
 
\hat{\zeta}^{+}\left( \lambda _{n}^{\ast },x\right) \right) ^{\ast }dx.
 
\nonumber
 
\end{matrix}</math></center>
 
The two integrals in (20) are considered separately. The
 
first is
 
<center><math>\begin{matrix}
 
&&\int_{-b}^{b}\partial _{x}\phi ^{+}\left( \lambda _{m},x\right) \left(
 
\partial _{x}\hat{\phi}^{+}\left( \lambda _{n}^{\ast },x\right) \right)
 
^{\ast }dx \\
 
&=&\int_{-b}^{b}\left( \sum_{j=1}^{6}\mu _{j}\left( \lambda _{m}\right)
 
\alpha _{j}e^{\mu _{j}\left( \lambda _{m}\right) x}\right) \left(
 
\sum_{k=1}^{6}\mu _{k}\left( \lambda _{m}\right) \alpha _{k}e^{\mu
 
_{k}\left( \lambda _{n}\right) x}\right) dx \\
 
&=&\sum_{j=1}^{6}\sum_{k=1}^{6}\int_{-b}^{b}-\mu _{j}\left( \lambda
 
_{m}\right) ^{2}\alpha _{j}e^{\mu _{j}\left( \lambda _{m}\right) x}\alpha
 
_{k}e^{\mu _{k}\left( \lambda _{n}\right) x}dx \\
 
&=&\sum_{j=1}^{6}\sum_{k=1}^{6}-\mu _{j}\left( \lambda _{m}\right)
 
^{2}\alpha _{j}\alpha _{k}\left( \frac{e^{\left( \mu _{j}\left( \lambda
 
_{m}\right) +\mu _{k}\left( \lambda _{n}\right) \right) b}-e^{-\left( \mu
 
_{j}\left( \lambda _{m}\right) +\mu _{k}\left( \lambda _{n}\right) \right) b}
 
}{\mu _{j}\left( \lambda _{m}\right) +\mu _{k}\left( \lambda _{n}\right) }
 
\right)  \nonumber
 
\end{matrix}</math></center>
 
and the second is
 
<center><math>\begin{matrix}
 
&&\int_{-b}^{b}(1+P)i\zeta ^{+}\left( \lambda _{m}^{\ast },x\right) \left( i
 
\hat{\zeta}^{+}\left( \lambda _{n}^{\ast },x\right) \right) ^{\ast }dx \\
 
&=&\int_{-b}^{b}\left( \sum_{j=1}^{6}-\frac{\alpha _{j}}{\lambda _{m}}
 
\left( \mu _{j}\left( \lambda _{m}\right) ^{2}+\beta \mu _{j}\left( \lambda
 
_{m}\right) ^{6}\right) e^{\mu _{j}\left( \lambda _{m}\right) x}\right)
 
\left( \sum_{k=1}^{6}-\frac{\alpha _{k}}{\lambda _{n}}\mu _{k}\left(
 
\lambda _{n}\right) ^{2}e^{\mu _{k}\left( \lambda _{n}\right) x}\right) dx \\
 
&=&\sum_{j=1}^{6}\sum_{k=1}^{6}\int_{-b}^{b}\frac{\alpha _{j}\alpha _{k}}{
 
\lambda _{m}\lambda _{n}}\left( \mu _{j}\left( \lambda _{m}\right)
 
^{2}+\beta \mu _{j}\left( \lambda _{m}\right) ^{6}\right) \mu _{k}\left(
 
\lambda _{n}\right) ^{2}e^{\mu _{j}\left( \lambda _{m}\right) x}e^{\mu
 
_{k}\left( \lambda _{n}\right) x}dx \\
 
&=&\sum_{j=1}^{6}\sum_{k=1}^{6}\frac{\alpha _{j}\alpha _{k}}{\lambda
 
_{m}\lambda _{n}}\left( \mu _{j}\left( \lambda _{m}\right) ^{2}+\beta \mu
 
_{j}\left( \lambda _{m}\right) ^{6}\right) \mu _{k}\left( \lambda
 
_{n}\right) ^{2}\left( \frac{e^{\left( \mu _{j}\left( \lambda _{m}\right)
 
+\mu _{k}\left( \lambda _{n}\right) \right) b}-e^{-\left( \mu _{j}\left(
 
\lambda _{m}\right) +\mu _{k}\left( \lambda _{n}\right) \right) b}}{\mu
 
_{j}\left( \lambda _{m}\right) +\mu _{k}\left( \lambda _{n}\right) }\right)
 
\\
 
&=&\sum_{j=1}^{6}\sum_{k=1}^{6}\frac{\alpha _{j}\alpha _{k}}{\lambda
 
_{m}\lambda _{n}}\left( -\lambda _{m}^{2}\right) \mu _{k}\left( \lambda
 
_{n}\right) ^{2}\left( \frac{e^{\left( \mu _{j}\left( \lambda _{m}\right)
 
+\mu _{k}\left( \lambda _{n}\right) \right) b}-e^{-\left( \mu _{j}\left(
 
\lambda _{m}\right) +\mu _{k}\left( \lambda _{n}\right) \right) b}}{\mu
 
_{j}\left( \lambda _{m}\right) +\mu _{k}\left( \lambda _{n}\right) }\right) .
 
\nonumber
 
\end{matrix}</math></center>
 
Therefore the calculation of the inner product in equation (\ref
 
{innerprodall}) does not require numerical integration.
 
 
===Solution===
 
 
By solving (18) using the eigenfunctions of <math>\mathcal{B}</math>
 
and <math>\mathcal{B}^{\ast }</math> we find the evolution of the plate, from an
 
initial displacement <math>U_{0}(x),</math> is
 
<center><math>
 
U\left( x,t\right) =\sum_{n=-\infty }^{\infty }e^{i\lambda _{n}t}\frac{
 
\left\langle U_{0}\left( x\right) ,\hat{\Phi}\left( \lambda _{n},x\right)
 
\right\rangle _{\mathcal{H}}}{\left\langle \Phi \left( \lambda _{n},x\right)
 
,\hat{\Phi}\left( \lambda _{n},x\right) \right\rangle _{\mathcal{H}}}\Phi
 
\left( \lambda _{n},x\right) .  (21)
 
</math></center>
 
The inner product of <math>U_{0}</math> with the eigenfunction <math>\hat{\Phi}\left(
 
\lambda _{n},x\right) </math> is the only quantity left to compute in (\ref
 
{evolutionB}). This inner product is written
 
<center><math>
 
\left\langle U_{0}\left( x\right) ,\hat{\Phi}\left( \lambda _{n},x\right)
 
\right\rangle _{\mathcal{H}}=\sum\limits_{j=1}^{6}\left(
 
\begin{matrix}{c}
 
\alpha _{j}\dint_{-b}^{b}-\mu _{j}\left( \lambda _{n}\right) ^{2}e^{\mu
 
_{j}\left( \lambda _{n}\right) x}\phi _{0}\left( x\right) dx+\left. \alpha
 
_{j}\mu _{j}\left( \lambda _{n}\right) e^{\mu _{j}\left( \lambda _{n}\right)
 
x}\phi _{0}\left( x\right) \right| _{-b}^{b} \\
 
\alpha _{j}\lambda _{n}\dint_{-b}^{b}e^{\mu _{j}\left( \lambda _{n}\right)
 
x}i\zeta _{0}\left( x\right) dx
 
\end{matrix}
 
\right)  (22)
 
</math></center>
 
and the integrals must be evaluated by numerical integration. The solutions
 
calculated using the Lax-Phillips scattering theory are identical to those
 
found using the self-adjoint operator method and for this reason no further
 
figures are shown.\pagebreak
 
 
==Conclusions==
 
 
The spectral theory of a linear thin plate floating on shallow water has
 
been derived. Two spectral-theory solutions have been presented which
 
determine the time-dependent motion of the thin plate. The first method was
 
based on self-adjoint operator theory and the second method was based on the
 
Lax-Phillips scattering. The self-adjoint method solved both the wave
 
forcing and the free plate problem while the Lax-Phillips method only solved
 
for a free plate. The eigenfunctions for the self-adjoint method are
 
orthogonal and the eigenvalues are continuous and consist of all <math>\mathbb{R}
 
, </math> which makes the calculation of the eigenvalues trivial. The Lax-Phillips
 
method has discrete eigenvalues which must be calculated numerically and the
 
system of eigenfunctions is biorthogonal. The advantage of the Lax-Phillips
 
method is that the modes of vibration of the plate-water system and their
 
frequency and rate of decay are found. While the relative speeds of the two
 
methods depends of the exact way in which they are implemented, the
 
Lax-Phillips method should be considerably faster if the eigenvalues have
 
been determined.
 
 
The development of a spectral theory for more complicated hydroelastic
 
problems remains a major challenge. While this theory must be more
 
complicated than that presented here, many features can be expected to
 
remain. For example, [[ohmatsuVLFS]] has shown that the single frequency
 
solutions can be used to solve certain time-dependent problems and \cite
 
{Hazard} have shown that modes, in which the solution can be expanded, exist
 
for other hydroelastic systems.
 
 
{\Large Acknowledgments}
 
 
\begin{acknowledgment}
 
I would like to thank the anonymous reviewers, Dr. Kathy Ruggerio, and Prof.
 
James Sneyd for their very helpful comments. Also, Prof. Boris Pavlov for
 
explaining the Lax-Phillips scattering. \pagebreak
 
\end{acknowledgment}
 
 
\bibliographystyle{jfm}
 
\bibliography{mike,others}
 
\pagebreak
 
 
\begin{center}
 
{\huge Figure Captions}
 
\end{center}
 
 
\textsc{Figure} 1. Schematic diagram of a thin plate floating on shallow
 
water and the coordinates and dimensions of the problem.
 
 
\textsc{Figure} 2. The evolution of the displacement due to a pulse
 
travelling to the right for the times shown. The plate occupies the region <math>
 
-50\leq x\leq 50<math> and is shown by the bold line. </math>\beta =2\times
 
10^{4},b=50. </math>
 
 
\textsc{Figure} 3. The evolution of the potential due to a pulse travelling
 
to the right for the times shown. The plate occupies the region <math>-50\leq
 
x\leq 50<math> and is shown by the bold line. </math>\beta =2\times 10^{4},b=50.</math>
 
 
\textsc{Figure} 4. The evolution of the displacement for a plate released at
 
<math>t=0</math> for the times shown. The plate occupies the region <math>-50\leq x\leq 50</math>
 
and is shown by the bold line. <math>\beta =2\times 10^{4},b=50.</math>
 
 
\textsc{Figure} 5. The evolution of the displacement for a plate released at
 
<math>t=0</math> for the times shown. The plate occupies the region <math>-50\leq x\leq 50</math>
 
and is shown by the bold line. <math>\beta =2\times 10^{4},b=50.</math>
 
 
\textsc{Figure} 6. The location of the first 19 eigenvalues <math>\lambda _{n}</math>
 
of <math>\mathcal{B}</math> for <math>\beta =2\times 10^{4},\;b=50.</math>
 
 
\textsc{Figure} 7. The real (solid) and imaginary (dashed) parts of the
 
resonance eigenfunctions for <math>n=1,</math> <math>3</math>, 5, and <math>7</math> as shown. <math>\beta
 
=2\times 10^{4},\;b=50.</math>
 
 
 
\pagebreak
 
 
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Latest revision as of 11:04, 6 April 2008

Michael H Meylan, Spectral Solution of Time Dependent Shallow Water Hydroelasticity, J. of Fluid Mechanics, 454, pp 387-402, 2002.

Time-Dependent Linear Water Wave problem of a Floating Elastic Plate on Shallow Depth. The solution was found using a Generalised Eigenfunction Expansion and as a sum over Scattering Frequencies.