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| | pp 387-402, 2002. | | pp 387-402, 2002. |
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| − | [[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]] |
| − |
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| − |
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| − |
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| − | = 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>}
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| − | ?? and in revised form ??
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| − |
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| − |
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| − |
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| − | The spectral theory of a thin plate floating on shallow water is derived and
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| − | used to solve the time-dependent motion. This theory is based on an energy
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| − | inner product in which the evolution operator becomes unitary. Two solution
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| − | methods are presented. In the first, the solution is expanded in the
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| − | eigenfunctions of a self-adjoint operator, which are the incoming wave
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| − | solutions for a single frequency. In the second, the scattering theory of
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| − | Lax-Phillips is used. The Lax-Phillips scattering solution is suitable for
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| − | calculating only the free motion of the plate. However, it determines the
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| − | modes of vibration of the plate-water system. These modes, both oscillate
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| − | and decay, are found by a complex search algorithm based contour
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| − | integration. As well as an application to modelling floating runways, the
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| − | spectral-theory for a floating thin plate on shallow water is a solvable
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| − | model for more complicated hydroelastic systems.
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| − |
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| − |
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| − | ==Introduction==
| |
| − |
| |
| − | Hydroelasticity is the study of immersed or floating elastic bodies in a
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| − | fluid. It has a wide range of applications including very large floating
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| − | structures ([[Kashiwagihydro98]]), ships ([[Bishop]]), breakwaters (
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| − | [[Stoker]]) and sea ice ([[Squire_Review]]). One of the best studied
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| − | hydroelastic models is the linear floating thin plate ([[Newman_deform]],
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| − | [[ohmatsuVLFS]], [[Kagemoto97]], and [[Kashiwagihydro98]]) because
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| − | it models many physical systems, such as a floating runway or an ice floe (
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| − | [[jgrfloecirc]]).
| |
| − |
| |
| − | The time-dependence in linear hydroelastic problems is usually removed by
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| − | solving for a single frequency which we will refer to as the \emph{single
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| − | frequency solution}. The solution is normally found by expanding the elastic
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| − | body motion in the free modes of vibration and solving the fluid equations
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| − | using a Green function ([[Bishop]]). This is analogous to solving for a
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| − | rigid floating body using the six rigid modes ([[Sarp_Isa]]). While
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| − | alternative methods have been developed ([[Kashiwagibspline]], \cite
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| − | {ohmatsuVLFS}, and [[Kagemoto97]]), these are based on exploiting some
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| − | property such as a simple geometry or high relative stiffness.
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| − |
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| − | In contrast to the single frequency solutions, solving time-dependent linear
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| − | hydroelastic systems remains a major challenge. [[Kashiwagitime]] and
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| − | [[Endotime]] have developed a time-stepping procedure; however, this
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| − | method results in error growth in time. Since the problem is linear it is
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| − | solvable by a spectral method which eliminates the long-time growth of
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| − | errors. Furthermore, such a method provides information about the behaviour
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| − | of the solution, such as the decay constant of the motion. However, the
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| − | spectral theory for linear hydroelasticity has not been developed. For this
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| − | reason, spectral type solutions such as [[Ohmatsutimedep]], based on a
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| − | Fourier expansion of the single frequency solution, have only solved the
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| − | problem in restricted circumstances.
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| − |
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| − | A floating thin plate on shallow water is considered in this paper. This
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| − | problem has been chosen for the following reasons: while the single
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| − | frequency solution is straightforward ([[Stoker]]), the time-dependent
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| − | solution has never been calculated; recently [[OhkusuNamba]], \cite
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| − | {OhkusuISOPE}, and [[Ertekinshallow1999]] used it to model a floating
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| − | runway; the spectral-theory developed here is a solvable model for more
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| − | complex hydroelastic systems.
| |
| − |
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| − | The spectral theory for a thin plate on shallow water is based on an inner
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| − | product which gives the energy of the plate-water system. With respect to
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| − | this inner product the evolution operator becomes unitary. Two different
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| − | solution methods are derived from this spectral theory. The first method is
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| − | based on an expansion of the solution in eigenfunctions of a self-adjoint
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| − | operator. These are the single frequency solutions. The second solution
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| − | method is based on the scattering theory of Lax-Phillips ([[laxphilips]]
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| − | ). It provides the solution in terms of a countable number of modes which
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| − | have both an oscillation and a decay. These modes are important as they can
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| − | be used to characterise the response of the system. With the exception of
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| − | [[Hazard]], they have not been investigated for hydroelastic systems.
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| − |
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| − | ==Formulation: A Thin Plate on Shallow Water==
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| − |
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| − | Figure \ref{shallow} shows a schematic diagram of the problem. The plate is
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| − | infinite in the <math>y</math> direction, so that only the <math>x</math> and <math>z</math> directions are
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| − | considered. The <math>x</math> direction is horizontal, the positive <math>z</math> axis points
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| − | vertically up, and the plate covers the region <math>-b\leqslant x\leqslant b.</math>
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| − | The water is of uniform depth <math>h</math> which is small<math>\ </math>enough that the water
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| − | may be approximated as shallow ([[Stoker]]). The amplitudes are assumed
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| − | small enough that the linear theory is appropriate, and the plate is
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| − | sufficiently thin that the shallow draft approximation may be made (\cite
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| − | {OhkusuISOPE}). The solution could be extended to waves incident at an angle
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| − | on a infinite two dimensional plate, as described in [[OhkusuISOPE]], but
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| − | to keep the treatment straightforward this is not done here.
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| − |
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| − | The mathematical description of the problem follows from [[Stoker]]. The
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| − | kinematic condition is
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| − | <center><math>
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| − | \partial _{t}\zeta =-h\partial _{x}^{2}\phi , (1)
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| − | </math></center>
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| − | 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
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| − | (from the shallow draft approximation). The equation derived by equating the
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| − | pressure at the free surface is
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| − | <center><math>
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| − | -\rho g\zeta -\rho \partial _{t}\phi =\left\{
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| − | \begin{matrix}{c}
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| − | 0,\;\;x\notin (-b,b), \\
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| − | D\partial _{x}^{4}\zeta +\rho ^{\prime }d\partial _{t}^{2}\zeta ,\;\;x\in
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| − | (-b,b),
| |
| − | \end{matrix}
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| − | \right. (2)
| |
| − | </math></center>
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| − | where <math>D</math> is the bending rigidity of the plate per unit length, <math>\rho </math> is
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| − | the density of water, <math>\rho ^{\prime }</math> is the average density of the plate,
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| − | <math>g</math> is the acceleration due to gravity, and <math>d</math> is the thickness of the plate
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| − | <math>.</math> At the ends of the plate the free edge boundary conditions
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| − | <center><math>
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| − | \lim_{x\downarrow -b}\partial _{x}^{2}\zeta =\lim_{x\uparrow b}\partial
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| − | _{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
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| − | 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>
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| − | <center><math>
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| − | \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
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| − | b}\partial _{x}^{5}\phi =0 (4)
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| − | </math></center>
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| − | which will be used subsequently.
| |
| − |
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| − | Non-dimensional variables are now introduced. The space variables are
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| − | non-dimensionalised using the water depth <math>h,</math> and the time variables are
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| − | non-dimensionalised using <math>\sqrt{h/g}</math>. The non-dimensional variables are
| |
| − | <center><math>
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| − | \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}}.
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| − | </math></center>
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| − | In these new variables, (1) and (2) become
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| − | <center><math>
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| − | \partial _{\bar{t}}\bar{\zeta}=-\partial _{\bar{x}}^{2}\bar{\phi}
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| − | (5)
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| − | </math></center>
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| − | and
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| − | <center><math>
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| − | -\bar{\zeta}-\partial _{\bar{t}}\bar{\phi}=\left\{
| |
| − | \begin{matrix}{c}
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| − | 0,\;\;\bar{x}\notin (-\bar{b},\bar{b}), \\
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| − | \beta \partial _{\bar{x}}^{4}\bar{\zeta}+\gamma \partial _{\bar{t}}^{2}\bar{
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| − | \zeta},\;\;\bar{x}\in (-\bar{b},\bar{b}),
| |
| − | \end{matrix}
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| − | \right. (6)
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| − | </math></center>
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| − | where <math>\beta </math> and <math>\gamma </math> are
| |
| − | <center><math>
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| − | \beta =\frac{D}{\rho gh^{4}}\;\;\mathrm{and\ \ }\gamma =\frac{\rho ^{\prime
| |
| − | }d}{\rho h}.
| |
| − | </math></center>
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| − | For clarity the overbar is dropped from now on.
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| − |
| |
| − | The main change in extending the formulation to water of finite depth is
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| − | that the velocity potential will be governed by Laplace's equation. This
| |
| − | makes the solution of the problem much more computationally demanding since
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| − | 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.===
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| − |
| |
| − | 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.
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| − |
| |
| − | 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.
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| − |
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| − | ==The Energy Inner Product==
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| − |
| |
| − | 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}
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| − | \phi (x,t) \\
| |
| − | i\zeta (x,t)
| |
| − | \end{matrix}
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| − | \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
| |
| − |
| |
| − | \FRAME{ftFU}{5.5434in}{1.6544in}{0pt}{\Qcb{}}{\Qlb{shallow}}{shallow.eps}{
| |
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| |
| − |
| |
| − | Some nonsense\pagebreak
| |
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| |
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| − |
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| − |
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