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| | a [[Three-Dimensional Floating Elastic Plate]] | | a [[Three-Dimensional Floating Elastic Plate]] |
| | | | |
| − | = Two Dimensional Problem =
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| − |
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| − | == Equations of Motion ==
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| − |
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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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| − |
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| − |
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| − | For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given
| |
| − | by the following
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| − |
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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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| − |
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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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| − |
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| − | The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
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| − |
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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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| − |
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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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| − |
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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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| − |
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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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| − |
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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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| − |
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| − | and the kinematic condition
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| − |
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| − | <math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>
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| − |
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| − |
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| − |
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| − | The equation within the fluid is governed by [[Laplace's Equation]]
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| − |
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| − | <math>\nabla^2\phi =0 </math>
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| − |
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| − | and we have the no-flow condition through the bottom boundary
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| − |
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| − | <math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
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| − |
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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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| − |
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| − | == Solution Methods ==
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| − |
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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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| − |
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| − | === Single Crack ===
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| − |
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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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| − |
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| − | We present here the solution of [[Evans and Porter 2005]] for the simple
| |
| − | case of a single crack with waves incident from normal (they also considered multiple cracks
| |
| − | and waves incident from different angles).
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| − | The solution of [[Evans and Porter 2005]] expresses the potential
| |
| − | <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]]
| |
| − | given by
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| − |
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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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| − |
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| − | where
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| − |
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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),
| |
| − | </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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| − |
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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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| − |
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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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| − |
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| − | It can easily be shown that <math>\psi_s</math> is symmetric about <math>x = 0</math> and
| |
| − | <math>\psi_a</math> is antisymmetric about <math>x = 0</math>.
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| − |
| |
| − | Substituting (1) into (2) gives
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| − |
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| − | <math>
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| − | \psi_s(x,z)=
| |
| − | {
| |
| − | -\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)=
| |
| − | {
| |
| − | {\rm sgn}(x) i\frac{\beta}{\alpha}\sum_{n=-2}^\infty
| |
| − | \frac{g_n'\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|}},
| |
| − | </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}.
| |
| − | </math>
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| − |
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| − | We then express the solution to the problem as a linear combination of the
| |
| − | incident wave and pairs of source functions at each crack,
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| − |
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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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| − |
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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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| − |
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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.
| |
| − | </math>
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| − |
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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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| − |
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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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| − |
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| − | and
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| − |
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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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| − |
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| − | === Two Semi-Infinite Plates of Different Properties ===
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| − |
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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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| − |
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| − | ====[[Wiener-Hopf]]====
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| − |
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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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| − |
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| − | ====[[Eigenfunction Matching Method]]====
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| − |
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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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| − |
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| − | ====[[Residue Calculus]]====
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| − |
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| − | = Three Dimensional Problem =
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| − |
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| − | == Equations of Motion ==
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| − |
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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
| |
| − | </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,
| |
| − | </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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| − |
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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
| |
| − | </math></center>
| |
| − | where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
| |
| − | of the water, and <math>g</math> is the acceleration due to gravity.
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| − |
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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>
| |
| − | 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>.
| |
| − | In the non-dimensional variables equations ((plate)) and ((pressure)
| |
| − | ) 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)
| |
| − | </math></center>
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| − | where
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| − | <center><math>
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| − | \beta =\frac{D}{g\rho a^{4}}, \; \mathbf{and} \; \gamma =\frac{\rho _{i}h}{\rho
| |
| − | a}.
| |
| − | </math></center>
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| − | We shall refer to <math>\beta </math> and <math>\gamma </math> as the stiffness and mass
| |
| − | respectively.
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| − |
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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
| |
| − | 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>
| |
| − | \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{
| |
| − | \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
| |
| − | }\bar{t}}\right] .</math> In the complex variables the equation of motion of the
| |
| − | ice floe ((n-d_ice)) is
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| − | <center><math>
| |
| − | \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. (plate2)
| |
| − | </math></center>
| |
| − | From now on we will drop the overbar and assume all variables are
| |
| − | non-dimensional.
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| − |
| |
| − | ==Equations of Motion for the Water==
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| − |
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| − | We require the equation of motion for the water to solve equation (\ref
| |
| − | {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 , \\
| |
| − | {\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{
| |
| − | x}\notin \Delta ,
| |
| − | \end{matrix}
| |
| − | \right\} (bvp_nond)
| |
| − | </math></center>
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| − | ([[Weh_Lait]]). As before, <math>w</math> is the displacement of the floe and <math>p</math>
| |
| − | 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
| |
| − | occupied by the floe. The water is assumed infinitely deep. A schematic
| |
| − | diagram of this problem is shown in Figure (vibration).
| |
| − | \begin{figure}[tbp]
| |
| − | \begin{center}
| |
| − | \epsfbox{vibration.eps}
| |
| − | \end{center}
| |
| − | \caption{{The schematic diagram of the boundary value problem and the
| |
| − | coordinate system used in the solution.}}
| |
| − | (vibration)
| |
| − | \end{figure}
| |
| − |
| |
| − | The boundary value problem ((bvp)) is subject to an incident wave which
| |
| − | is imposed through a boundary condition as <math>\left| \mathbf{x}\right|
| |
| − | \rightarrow \infty </math>. This boundary condition, which is called the
| |
| − | Sommerfeld radiation condition, is essentially that at large distances the
| |
| − | potential consists of a radial outgoing wave (the wave generated by the ice
| |
| − | floe motion) and the incident wave. It is expressed mathematically as
| |
| − | <center><math>
| |
| − | \lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
| |
| − | \frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
| |
| − | \mathrm{In}})=0, (summerfield)
| |
| − | </math></center>
| |
| − | [[Weh_Lait]]. The incident potential (i.e. the incoming wave) <math>\phi ^{
| |
| − | \mathrm{In}}</math> is
| |
| − | <center><math>
| |
| − | \phi ^{\mathrm{In}}(x,y,z)=\frac{A}{\sqrt{\alpha }}e^{i\alpha (x\cos \theta
| |
| − | +y\sin \theta )}e^{\alpha z}, (input)
| |
| − | </math></center>
| |
| − | where <math>A</math> is the non-dimensional wave amplitude.
| |
| − |
| |
| − | The standard solution method to the linear wave problem is to transform the
| |
| − | boundary value problem into an integral equation using a Green function
| |
| − | \citep{john1,
| |
| − | john2,Sarp_Isa,jgrfloecirc}. Performing such a transformation, the boundary
| |
| − | value problem ((bvp)) and ((summerfield)) becomes
| |
| − | <center><math>
| |
| − | \phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
| |
| − | ;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
| |
| − | )\right) dS_{\mathbf{y}}. (water)
| |
| − | </math></center>
| |
| − | The Green function <math>G_{\alpha }</math> is
| |
| − | <center><math>
| |
| − | G_{\alpha }(\mathbf{x};\mathbf{y)}=\frac{1}{4\pi }\left( \frac{2}{|\mathbf{x}
| |
| − | -\mathbf{y}|}-\pi \alpha \left( \mathbf{H_{0}}(\alpha |\mathbf{x}-\mathbf{y}
| |
| − | |)+Y_{0}(\alpha |\mathbf{x}-\mathbf{y}|)\right) +2\pi i\alpha J_{0}(\alpha |
| |
| − | \mathbf{x}-\mathbf{y}|)\right) ,
| |
| − | </math></center>
| |
| − | [[Weh_Lait,jgrfloecirc]], where <math>J_{0}</math> and <math>Y_{0}</math> are respectively
| |
| − | Bessel functions of the first and second kind of order zero, and <math>\mathbf{
| |
| − | H_{0}}</math> is the Struve function of order zero [[abr_ste]]. A solution for
| |
| − | water of finite depth could be found by simply using the depth dependent
| |
| − | Green function [[Weh_Lait]].
| |
| − |
| |
| − | The integral equation ((water)) will be solved using numerical
| |
| − | integration. The only difficulty arises from the non-trivial nature of the
| |
| − | kernel of the integral equation (the Green function). However, the Green
| |
| − | function has no <math>z</math> dependence due to the shallow draft approximation and
| |
| − | depends only on <math>|\mathbf{x}-\mathbf{y}|.</math> This means that the Green
| |
| − | function is one dimensional and the values which are required for a given
| |
| − | calculation can be looked up in a previously computed table.
| |
| − |
| |
| − | ==Solving for the Wave Induced Ice Floe Motion==
| |
| − |
| |
| − | To determine the ice floe motion we must solve equations ((plate2)) and (
| |
| − | (water)) simultaneously. We do this by expanding the floe motion in the
| |
| − | free modes of vibration of a thin plate. The major difficulty with this
| |
| − | method is that the free modes of vibration can be determined analytically
| |
| − | only for very restrictive geometries, e.g. a circular thin plate. Even the
| |
| − | free modes of vibration of a square plate with free edges must be determined
| |
| − | numerically. This is the reason why the solution of [[jgrfloecirc]] was
| |
| − | only for a circular floe.
| |
| − |
| |
| − | Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
| |
| − | conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
| |
| − | which satisfy the free boundary conditions and the following eigenvalue
| |
| − | equation
| |
| − | <center><math>
| |
| − | \nabla ^{4}w_{i}=\lambda _{i}w_{i}.
| |
| − | </math></center>
| |
| − | The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
| |
| − | orthogonal and the eigenvalues are positive and real. While the plate will
| |
| − | always have repeated eigenvalues, orthogonal modes can still be found and
| |
| − | the modes can be normalized. We therefore assume that the modes are
| |
| − | orthonormal, i.e.
| |
| − | <center><math>
| |
| − | \iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
| |
| − | dS_{\mathbf{Q}}=\delta _{ij}
| |
| − | </math></center>
| |
| − | where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
| |
| − | have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
| |
| − | \infty </math> and we order the modes by increasing eigenvalue. These modes can be
| |
| − | used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
| |
| − | .
| |
| − |
| |
| − | We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
| |
| − | <center><math>
| |
| − | w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
| |
| − | (expansion)
| |
| − | </math></center>
| |
| − | From the linearity of ((water)) the potential can be written in the
| |
| − | following form
| |
| − | <center><math>
| |
| − | \phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
| |
| − | </math></center>
| |
| − | where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
| |
| − | <center><math>
| |
| − | \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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| − |
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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(
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| − | \mathbf{Q}\right) dS_{\mathbf{Q}} (final)
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| − | </math></center>
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| − | which is a matrix equation in <math>c_{i}.</math>
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| − |
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| − | We cannot solve equation ((final)) without determining the modes of
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| − | vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
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| − | \lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
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| − | {phii}). We use the finite element method to determine the modes of
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| − | vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
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| − | (phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
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| − | set of nodes is used for the finite element method and to define the panels
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| − | for the integral equation.
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| | | | |
| | [[Category:Linear Hydroelasticity]] | | [[Category:Linear Hydroelasticity]] |
