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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_1957a|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_1963a|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_Dixon_2000A|Squire and Dixon 2000]] and [[Williams_Squire_2002A|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_Dixon_2001a|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_Porter_2005a|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_Porter_2005a|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_Porter_2005a|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_Davies_1968a|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_Squire_1994a|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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| − | | |
| − | <math>D\nabla ^4 w + \rho _i h w = p</math>
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| − | | |
| − | ==The Equation of Motion for the Ice Floe==
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| − | Ice floes range in size from much smaller to much larger than the dominant
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| − | | |
| − | wavelength of the ocean waves. However there are two reasons why solutions
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| − | for ice floes of intermediate size (a size similar to the wavelength) are
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| − | the most important. The first is that at these intermediate sizes ice floes
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| − | | |
| − | scatter significant wave energy. The second is that, since it is wave
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| − | induced flexure which determines the size of ice floes in the MIZ, ice floes
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| − | tend to form most often at this intermediate length.
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| − | The theory for an ice floe of intermediate size which is developed in this
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| − | paper obviously also applies to small or large floes. However, if the
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| − | solution for a small or large floe is required then the appropriate simpler
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| − | | |
| − | theory should be used. Small ice floes (ice floes much small than the
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| − | | |
| − | wavelength) should be modelled as rigid \citep{Masson_Le,Massondrift}. Large
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| − | ice floes (ice floes much larger than the wavelength) should be modelled as
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| − | infinite and flexible \citep{FoxandSquire}. In the intermediate region,
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| − | where the size of the wavelength is similar to the size of the ice floe, the
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| − | ice floe must be modelled as finite and flexible.
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| − | We model the ice floe as a thin plate of constant thickness and shallow
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| − | draft following \citet{Wadhams1986} and \citet{Squire_Review}. The thin
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| − | plate equation \citep{Hildebrand65} gives the following equation of motion
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| − | for the ice floe
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| − | <center><math>
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| − | D\nabla ^{4}W+\rho _{i}h\frac{\partial ^{2}W}{\partial t^{2}}=p,
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| − | | |
| − | \label{plate}
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| − | | |
| − | </math></center>
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| − | where <math>W</math> is the floe displacement, <math>\rho _{i}</math> is the floe density, <math>h</math> is
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| − | | |
| − | the floe thickness, <math>p</math> is the pressure, and <math>D</math> is the modulus of rigidity
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| − | | |
| − | of the floe (<math>D=Eh^{3}/12(1-\nu ^{2})</math> where <math>E</math> is the Young's modulus and <math>
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| − | \nu <math> is Poisson's ratio). Visco-elastic effects can be included by making </math>
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| − | D </math> have some imaginary (damping)\ component but this will not be done here
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| − | | |
| − | to keep the presented results simpler. We assume that the plate is in
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| − | contact with the water at all times so that the water surface displacement
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| − | | |
| − | is also <math>W.</math> Equation (\ref{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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| − | | |
| − | \dfrac{\partial ^{2}W}{\partial n^{2}}+\nu \dfrac{\partial ^{2}W}{\partial
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| − | | |
| − | s^{2}}=0,\;\;\;\text{\textrm{and}}\mathrm{\;\;\;}\dfrac{\partial ^{3}W}{
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| − | \partial n^{3}}+\left( 2-\nu \right) \dfrac{\partial ^{3}W}{\partial
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| − | n\partial s^{2}}=0, \label{boundaryplate}
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| − | | |
| − | </math></center>
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| − | \citep{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 \label{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>. The non-dimensional
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| − | | |
| − | variables, denoted by an overbar, are
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| − | | |
| − | <center><math>
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| − | | |
| − | \bar{x}=\frac{x}{a},\;\;\bar{y}=\frac{y}{a},\;\;\bar{z}=\frac{z}{a},\;\;\bar{
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| − | W}=\frac{W}{a},\;\;\bar{t}=t\sqrt{\frac{g}{a}},\;\;\text{and}\;\;\bar{\Phi}=
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| − | \frac{\Phi }{a\sqrt{ag}}.
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| − | | |
| − | </math></center>
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| − | In the non-dimensional variables equations (\ref{plate}) and (\ref{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}, \label{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}}\;\;\text{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})=\func{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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| − | \func{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 (\ref{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. \label{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> \label{bvp}
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| − | | |
| − | \left.
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| − | | |
| − | \begin{array}{rr}
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| − | | |
| − | \nabla ^{2}\phi =0, & -\infty <z<0, \\
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| − | | |
| − | {\dfrac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
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| − | | |
| − | {\dfrac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
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| − | \mathbf{x}\in \Delta , \\
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| − | | |
| − | {\dfrac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
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| − | x}\notin \Delta ,
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| − | \end{array}
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| − | \right\} \label{bvp_nond}
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| − | | |
| − | </math></center>
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| − | (\citet{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 \ref{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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| − | | |
| − | \label{vibration}
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| − | | |
| − | \end{figure}
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| − | | |
| − | | |
| − | | |
| − | The boundary value problem (\ref{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, \label{summerfield}
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| − | | |
| − | </math></center>
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| − | \citep{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}, \label{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 (\ref{bvp}) and (\ref{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}}. \label{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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| − | \citep{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 \citep{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 \citep{Weh_Lait}.
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| − | | |
| − | | |
| − | | |
| − | The integral equation (\ref{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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