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| = Introduction =
| | This page has moved to [[:Category:Floating Elastic Plate|Floating Elastic Plate]]. |
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| The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
| | Please change the link to the new page |
| physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
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| more than 100 years ago and a discussion of the problem appears in [[Stoker 1957]]. The problem can
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| be divided into the two and three dimensional formulations which are closely related. The plate is assumed to be isotropic while the water motion is irrotational and inviscid.
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| = Two Dimensional Problem =
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| == Equations of Motion ==
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| When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the [[Bernoulli-Euler Beam]] which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the [[Timoshenko Beam]] theory and [[Reddy-Bickford Beam]] theory where shear deformation of higher order is considered.
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| For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given
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| by the following
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| <math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>
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| where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the beam,
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| <math>h</math> is the thickness of the beam (assumed constant), <math> p</math> is the pressure
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| and <math>\eta</math> is the beam vertical displacement.
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| The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
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| <math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>
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| at the edges of the plate.
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| The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
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| pressure at the surface), i.e.
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| <math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>
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| where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
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| is the velocity potential. The velocity potential is governed by Laplace's equation through out
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| the fluid domain subject to the free surface condition and the condition of no flow through the
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| bottom surface. If we denote the beam-covered (or possible multiple beams covered) region of the fluid by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
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| [[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
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| [[Finite Depth]] are the following. At the surface
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| we have the dynamic condition
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| <math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
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| i\omega \rho \phi, \, z=0, \, x\in P</math>
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| <math>0=
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| \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>
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| and the kinematic condition
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| <math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>
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| The equation within the fluid is governed by [[Laplace's Equation]]
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| <math>\nabla^2\phi =0 </math>
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| and we have the no-flow condition through the bottom boundary
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| <math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
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| (so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
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| free surface or plate covered surface are at <math>z=0</math>).
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| <math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
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| are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
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| the thickness and flexural rigidity of the plate.
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| == Solution Methods ==
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| There are many different methods to solve the corresponding equations ranging from highly analytic such
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| as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
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| applicable and have advantages in different situations. We describe here some of the solutions
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| which have been developed grouped by problem
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| === Single Crack ===
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| The simplest problem to consider is one where there are only two semi-infinite plates of identical properties separated by a crack. A related problem in acoustics was considered by [[Kouzov 1963]] who used an integral representation of the problem to solve it explicitly using the Riemann-Hilbert technique. Recently the crack problem has been considered by [[Squire and Dixon 2000]] and [[Williams and Squire 2002]] using a Green function method applicable to infinitely deep water and they obtained simple expressions for the reflection and transmission coefficients. [[Squire and Dixon 2001]] extended the single crack problem to a multiple crack problem in which the semi-infinite regions are separated by a region consisting of a finite number of plates of finite size with all plates having identical properties. [[Evans and Porter 2005]] further considered the multiple crack problem for finitely deep water and provided an explicit solution.
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| We present here the solution of [[Evans and Porter 2005]] for the simple
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| case of a single crack with waves incident from normal (they also considered multiple cracks
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| and waves incident from different angles).
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| The solution of [[Evans and Porter 2005]] expresses the potential
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| <math>\phi</math> in terms of a linear combination of the incident wave and certain source functions located at the crack.
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| Along with satisfying the field and boundary conditions, these source functions satisfy the jump conditions in the displacements and gradients across the crack.
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| They first define <math>\chi(x,z)</math> to be the Two-Dimensional solution to the [[Free-Surface Green Function for a Floating Elastic Plate]]
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| given by
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| <math>
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| \chi(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k_n h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-k_n|x|},\,\,\,(1)
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| </math>
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| where
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| <math>
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| C_n=\frac{1}{2}\left(h + \frac{(5\beta k_n ^4 + 1 - \alpha\gamma)\sin^2{(k_n h)}}{\alpha}\right),
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| </math>
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| and <math>k_n</math> are the solutions of the [[Dispersion Relation for a Floating Elastic Plate]].
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| Consequently, the source functions for a single crack at <math>x=0</math> can be defined as
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| <math>
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| \psi_s(x,z)= \beta\chi_{xx}(x,z),\,\,\,
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| \psi_a(x,z)= \beta\chi_{xxx}(x,z),\,\,\,(2)
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| </math>
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| It can easily be shown that <math>\psi_s</math> is symmetric about <math>x = 0</math> and
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| <math>\psi_a</math> is antisymmetric about <math>x = 0</math>.
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| Substituting (1) into (2) gives
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| <math>
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| \psi_s(x,z)=
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| {
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| -\frac{\beta}{\alpha}
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| \sum_{n=-2}^\infty
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| \frac{g_n\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|} },
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| \psi_a(x,z)=
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| {
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| {\rm sgn}(x) i\frac{\beta}{\alpha}\sum_{n=-2}^\infty
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| \frac{g_n'\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|}},
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| </math>
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| where
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| <math>
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| g_n = ik_n^3 \sin{k_n h},\,\,\,\,
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| g'_n= -k_n^4 \sin{k_n h}.
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| </math>
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| We then express the solution to the problem as a linear combination of the
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| incident wave and pairs of source functions at each crack,
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| <math>
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| \phi(x,z) =
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| e^{-k_0 x}\frac{\cos(k_0(z+h))}{\cos(k_0h)}
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| + (P\psi_s(x,z) + Q\psi_a(x,z))\,\,\,(3)
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| </math>
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| where <math>P</math> and <math>Q</math> are coefficients to be solved which represent the jump in the gradient
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| and elevation respectively of the plates across the crack <math>x = a_j</math>.
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| The coefficients <math>P</math> and <math>Q</math> are found by applying the edge conditions and to
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| the <math>z</math> derivative of <math>\phi</math> at <math>z=0</math>,
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| <math>
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| \frac{\partial^2}{\partial x^2}\left. \frac{\partial \phi}{\partial z}\right|_{x=0,z=0}=0,\,\,\,
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| {\rm and}\,\,\,\,
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| \frac{\partial^3}{\partial x^3}\left. \frac{\partial \phi}{\partial z}\right|_{x=0,z=0}=0.
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| </math>
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| The reflection and transmission coefficients, <math>R</math> and <math>T</math> can be found from (3)
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| by taking the limits as <math>x\rightarrow\pm\infty</math> to obtain
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| <math>
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| R = {- \frac{\beta}{\alpha}
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| (g'_0Q + ig_0P)}
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| </math>
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| and
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| <math>
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| T= 1 + {\frac{\beta}{\alpha}(g'_0Q - ig_0P)}
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| </math>
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| === Two Semi-Infinite Plates of Different Properties ===
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| The next most simple problem is two semi-infinite plates of different properties. Often one of
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| the plates is taken to be open water which makes the problem simpler. In general, the solution method
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| developed for open water can be extended to two plates of different properties, the exception to
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| this is the [[Residue Calculus]] solution which applies only when one of the semi-infinite regions
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| is water.
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| ====[[Wiener-Hopf]]====
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| The solution to the problem of two semi-infinite plates with different properties can be
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| solved by the Wiener-Hopf method. The first work on this problem was by [[Evans and Davies 1968]]
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| but they did not actually develop the method sufficiently to be able to calculate the solution.
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| The explicit solution was not found until the work of ...
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| ====[[Eigenfunction Matching Method]]====
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| The eigenfunction matching solution was developed by [[Fox and Squire 1994]].
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| Essentially the solution is expanded on either side of the crack.
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| ====[[Residue Calculus]]====
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| = Three Dimensional Problem =
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| == Equations of Motion ==
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| For a classical thin plate, the equation of motion is given by
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| <center><math>
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| D\nabla ^4 w + \rho _i h w = p
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| </math></center>
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| Equation ((plate)) is subject to the free edge boundary
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| conditions for a thin plate
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| <center><math>
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| \frac{\partial ^{2}w}{\partial n^{2}}+\nu \frac{\partial ^{2}w}{\partial
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| s^{2}}=0,\;\;\;=\textrm{and= }\mathrm{\;\;\;}\frac{\partial ^{3}w}{
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| \partial n^{3}}+\left( 2-\nu \right) \frac{\partial ^{3}w}{\partial
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| n\partial s^{2}}=0, (boundaryplate)
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| </math></center>
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| [[Hildebrand65]] where <math>n</math> and <math>s</math> denote the normal and tangential
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| directions respectively.
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| The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
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| water surface,
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| <center><math>
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| p=-\rho \frac{\partial \phi }{\partial t}-\rho gW (pressure)
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| </math></center>
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| where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
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| of the water, and <math>g</math> is the acceleration due to gravity.
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| We now introduce non-dimensional variables. We non-dimensionalise the length
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| variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
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| We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math> and
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| the mass variables with respect to <math>\rho a^{3}</math>.
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| In the non-dimensional variables equations ((plate)) and ((pressure)
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| ) become
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| <center><math>
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| \beta \nabla ^{4}\bar{w}+\gamma \frac{\partial ^{2}\bar{w}}{\partial \bar{t}
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| ^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{w}, (n-d_ice)
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| </math></center>
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| where
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| <center><math>
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| \beta =\frac{D}{g\rho a^{4}}\;\;\{mathrm and}\ \ = \gamma =\frac{\rho _{i}h}{\rho
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| a}.
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| </math></center>
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| We shall refer to <math>\beta </math> and <math>\gamma </math> as the stiffness and mass
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| respectively.
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| We will determine the response of the ice floe to wave forcing of a single
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| frequency (the response for more complex wave forcing can be found by
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| superposition of the single frequency solutions). Since the equations of
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| motion are linear the displacement and potential must have the same single
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| frequency dependence. Therefore they can be expressed as the real part of a
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| complex quantity whose time dependence is <math>e^{-i\sqrt{\alpha }t}</math> where <math>
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| \alpha <math> is the non-dimensional wavenumber and we write </math>\bar{W}(\bar{x},
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| \bar{y},\bar{t})={Re}\left[ w\left( \bar{x},\bar{y}\right) e^{-i\sqrt{
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| \alpha }\bar{t}}\right] \ <math>and</math>\;\Phi (\bar{x},\bar{y},\bar{z},\bar{t})=
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| {Re}\left[ \phi \left( \bar{x},\bar{y},\bar{z}\right) e^{-i\sqrt{\alpha
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| }\bar{t}}\right] .</math> In the complex variables the equation of motion of the
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| ice floe ((n-d_ice)) is
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| <center><math>
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| \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. (plate2)
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| </math></center>
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| From now on we will drop the overbar and assume all variables are
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| non-dimensional.
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| ==Equations of Motion for the Water==
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| We require the equation of motion for the water to solve equation (\ref
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| {plate2}). We begin with the non-dimensional equations of potential theory
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| which describe linear surface gravity waves
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| <center><math> (bvp)
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| \left.
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| \begin{matrix}{rr}
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| \nabla ^{2}\phi =0, & -\infty <z<0, \\
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| {\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
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| {\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
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| \mathbf{x}\in \Delta , \\
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| {\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
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| x}\notin \Delta ,
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| \end{matrix}
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| \right\} (bvp_nond)
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| </math></center>
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| ([[Weh_Lait]]). As before, <math>w</math> is the displacement of the floe and <math>p</math>
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| is the pressure at the water surface. The vector <math>\mathbf{x=(}x,y)</math> is a
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| point on the water surface and <math>\Delta </math> is the region of the water surface
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| occupied by the floe. The water is assumed infinitely deep. A schematic
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| diagram of this problem is shown in Figure (vibration).
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| \begin{figure}[tbp]
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| \begin{center}
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| \epsfbox{vibration.eps}
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| \end{center}
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| \caption{{The schematic diagram of the boundary value problem and the
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| coordinate system used in the solution.}}
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| (vibration)
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| \end{figure}
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| The boundary value problem ((bvp)) is subject to an incident wave which
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| is imposed through a boundary condition as <math>\left| \mathbf{x}\right|
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| \rightarrow \infty </math>. This boundary condition, which is called the
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| Sommerfeld radiation condition, is essentially that at large distances the
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| potential consists of a radial outgoing wave (the wave generated by the ice
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| floe motion) and the incident wave. It is expressed mathematically as
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| <center><math>
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| \lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
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| \frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
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| \mathrm{In}})=0, (summerfield)
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| </math></center>
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| [[Weh_Lait]]. The incident potential (i.e. the incoming wave) <math>\phi ^{
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| \mathrm{In}}</math> is
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| <center><math>
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| \phi ^{\mathrm{In}}(x,y,z)=\frac{A}{\sqrt{\alpha }}e^{i\alpha (x\cos \theta
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| +y\sin \theta )}e^{\alpha z}, (input)
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| </math></center>
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| where <math>A</math> is the non-dimensional wave amplitude.
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| The standard solution method to the linear wave problem is to transform the
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| boundary value problem into an integral equation using a Green function
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| \citep{john1,
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| john2,Sarp_Isa,jgrfloecirc}. Performing such a transformation, the boundary
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| value problem ((bvp)) and ((summerfield)) becomes
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| <center><math>
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| \phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
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| ;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
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| )\right) dS_{\mathbf{y}}. (water)
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| </math></center>
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| The Green function <math>G_{\alpha }</math> is
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| <center><math>
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| G_{\alpha }(\mathbf{x};\mathbf{y)}=\frac{1}{4\pi }\left( \frac{2}{|\mathbf{x}
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| -\mathbf{y}|}-\pi \alpha \left( \mathbf{H_{0}}(\alpha |\mathbf{x}-\mathbf{y}
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| |)+Y_{0}(\alpha |\mathbf{x}-\mathbf{y}|)\right) +2\pi i\alpha J_{0}(\alpha |
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| \mathbf{x}-\mathbf{y}|)\right) ,
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| </math></center>
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| [[Weh_Lait,jgrfloecirc]], where <math>J_{0}</math> and <math>Y_{0}</math> are respectively
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| Bessel functions of the first and second kind of order zero, and <math>\mathbf{
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| H_{0}}</math> is the Struve function of order zero [[abr_ste]]. A solution for
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| water of finite depth could be found by simply using the depth dependent
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| Green function [[Weh_Lait]].
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| The integral equation ((water)) will be solved using numerical
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| integration. The only difficulty arises from the non-trivial nature of the
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| kernel of the integral equation (the Green function). However, the Green
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| function has no <math>z</math> dependence due to the shallow draft approximation and
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| depends only on <math>|\mathbf{x}-\mathbf{y}|.</math> This means that the Green
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| function is one dimensional and the values which are required for a given
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| calculation can be looked up in a previously computed table.
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| ==Solving for the Wave Induced Ice Floe Motion==
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| To determine the ice floe motion we must solve equations ((plate2)) and (
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| (water)) simultaneously. We do this by expanding the floe motion in the
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| free modes of vibration of a thin plate. The major difficulty with this
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| method is that the free modes of vibration can be determined analytically
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| only for very restrictive geometries, e.g. a circular thin plate. Even the
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| free modes of vibration of a square plate with free edges must be determined
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| numerically. This is the reason why the solution of [[jgrfloecirc]] was
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| only for a circular floe.
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| Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
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| conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
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| which satisfy the free boundary conditions and the following eigenvalue
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| equation
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| <center><math>
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| \nabla ^{4}w_{i}=\lambda _{i}w_{i}.
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| </math></center>
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| The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
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| orthogonal and the eigenvalues are positive and real. While the plate will
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| always have repeated eigenvalues, orthogonal modes can still be found and
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| the modes can be normalized. We therefore assume that the modes are
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| orthonormal, i.e.
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| <center><math>
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| \iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
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| dS_{\mathbf{Q}}=\delta _{ij}
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| </math></center>
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| where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
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| have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
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| \infty </math> and we order the modes by increasing eigenvalue. These modes can be
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| used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
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| .
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| We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
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| <center><math>
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| w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
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| (expansion)
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| </math></center>
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| From the linearity of ((water)) the potential can be written in the
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| following form
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| <center><math>
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| \phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
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| </math></center>
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| where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
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| <center><math>
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| \phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
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| G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
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| (phi0)
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| </math></center>
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| and
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| <center><math>
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| \phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
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| )\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
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| dS_{\mathbf{y}}. (phii)
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| </math></center>
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| The potential <math>\phi _{0}</math> represents the potential due the incoming wave
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| assuming that the displacement of the ice floe is zero. The potentials <math>\phi
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| _{i}</math> represent the potential which is generated by the plate vibrating with
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| the <math>i</math>th mode in the absence of any input wave forcing.
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| We substitute equations ((expansion)) and ((expansionphi)) into
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| equation ((plate2)) to obtain
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| <center><math>
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| \beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
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| \sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
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| _{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}.
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| (expanded)
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| </math></center>
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| To solve equation ((expanded)) we multiply by <math>w_{j}</math> and integrate over
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| the plate (i.e. we take the inner product with respect to <math>w_{j})</math> taking
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| into account the orthogonality of the modes <math>w_{i}</math>, and obtain
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| <center><math>
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| \beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
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| \sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right)
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| +\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left(
| |
| \mathbf{Q}\right) dS_{\mathbf{Q}} (final)
| |
| </math></center>
| |
| which is a matrix equation in <math>c_{i}.</math>
| |
| | |
| We cannot solve equation ((final)) without determining the modes of
| |
| vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
| |
| \lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
| |
| {phii}). We use the finite element method to determine the modes of
| |
| vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
| |
| (phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
| |
| set of nodes is used for the finite element method and to define the panels
| |
| for the integral equation.
| |
| | |
| [[Category:Linear Hydroelasticity]]
| |
