|
|
| Line 10: |
Line 10: |
| | == Equations of Motion == | | == Equations of Motion == |
| | | | |
| − | 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. | + | 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. |
| | | | |
| | | | |
| Line 67: |
Line 67: |
| | the thickness and flexural rigidity of the plate. | | the thickness and flexural rigidity of the plate. |
| | | | |
| − | == Solution Methods == | + | == Solution Method == |
| | | | |
| | There are many different methods to solve the corresponding equations ranging from highly analytic such | | There are many different methods to solve the corresponding equations ranging from highly analytic such |
| | as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are | | as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are |
| − | applicable and have advantages in different situations. We describe here some of the solutions | + | applicable and have advantages in different situations. |
| − | which have been developed grouped by problem
| |
| − | | |
| − | === Single Crack ===
| |
| − | | |
| − | 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.
| |
| − | | |
| − | We present here the solution of [[Evans_Porter_2005a|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).
| |
| − | The solution of [[Evans_Porter_2005a|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.
| |
| − | Along with satisfying the field and boundary conditions, these source functions satisfy the jump conditions in the displacements and gradients across the crack.
| |
| − | 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
| |
| − | | |
| − | <math>
| |
| − | \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)
| |
| − | </math>
| |
| − | | |
| − | where
| |
| − | | |
| − | <math>
| |
| − | C_n=\frac{1}{2}\left(h + \frac{(5\beta k_n ^4 + 1 - \alpha\gamma)\sin^2{(k_n h)}}{\alpha}\right),
| |
| − | </math>
| |
| − | | |
| − | and <math>k_n</math> are the solutions of the [[Dispersion Relation for a Floating Elastic Plate]].
| |
| − | | |
| − | Consequently, the source functions for a single crack at <math>x=0</math> can be defined as
| |
| − | | |
| − | <math>
| |
| − | \psi_s(x,z)= \beta\chi_{xx}(x,z),\,\,\,
| |
| − | \psi_a(x,z)= \beta\chi_{xxx}(x,z),\,\,\,(2)
| |
| − | </math>
| |
| − | | |
| − | 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>.
| |
| − | | |
| − | Substituting (1) into (2) gives
| |
| − | | |
| − | <math>
| |
| − | \psi_s(x,z)=
| |
| − | {
| |
| − | -\frac{\beta}{\alpha}
| |
| − | \sum_{n=-2}^\infty
| |
| − | \frac{g_n\cos{(k_n(z+h))}}{2k_{xn}C_n}e^{k_n|x|} },
| |
| − | \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>
| |
| − | | |
| − | where
| |
| − | | |
| − | <math>
| |
| − | g_n = ik_n^3 \sin{k_n h},\,\,\,\,
| |
| − | g'_n= -k_n^4 \sin{k_n h}.
| |
| − | </math>
| |
| − | | |
| − | We then express the solution to the problem as a linear combination of the
| |
| − | incident wave and pairs of source functions at each crack,
| |
| − | | |
| − | <math>
| |
| − | \phi(x,z) =
| |
| − | e^{-k_0 x}\frac{\cos(k_0(z+h))}{\cos(k_0h)}
| |
| − | + (P\psi_s(x,z) + Q\psi_a(x,z))\,\,\,(3)
| |
| − | </math>
| |
| − | | |
| − | where <math>P</math> and <math>Q</math> are coefficients to be solved which represent the jump in the gradient
| |
| − | and elevation respectively of the plates across the crack <math>x = a_j</math>.
| |
| − | The coefficients <math>P</math> and <math>Q</math> are found by applying the edge conditions and to
| |
| − | the <math>z</math> derivative of <math>\phi</math> at <math>z=0</math>,
| |
| − | | |
| − | <math>
| |
| − | \frac{\partial^2}{\partial x^2}\left. \frac{\partial \phi}{\partial z}\right|_{x=0,z=0}=0,\,\,\,
| |
| − | {\rm and}\,\,\,\,
| |
| − | \frac{\partial^3}{\partial x^3}\left. \frac{\partial \phi}{\partial z}\right|_{x=0,z=0}=0.
| |
| − | </math>
| |
| − | | |
| − | The reflection and transmission coefficients, <math>R</math> and <math>T</math> can be found from (3)
| |
| − | by taking the limits as <math>x\rightarrow\pm\infty</math> to obtain
| |
| − | | |
| − | <math>
| |
| − | R = {- \frac{\beta}{\alpha}
| |
| − | (g'_0Q + ig_0P)}
| |
| − | </math>
| |
| − | | |
| − | and
| |
| − | | |
| − | <math>
| |
| − | T= 1 + {\frac{\beta}{\alpha}(g'_0Q - ig_0P)}
| |
| − | </math>
| |
| − | | |
| − | === Two Semi-Infinite Plates of Different Properties ===
| |
| − | | |
| − | The next most simple problem is two semi-infinite plates of different properties. Often one of
| |
| − | the plates is taken to be open water which makes the problem simpler. In general, the solution method
| |
| − | developed for open water can be extended to two plates of different properties, the exception to
| |
| − | this is the [[Residue Calculus]] solution which applies only when one of the semi-infinite regions
| |
| − | is water.
| |
| − | | |
| − | ====[[Wiener-Hopf]]====
| |
| − | | |
| − | The solution to the problem of two semi-infinite plates with different properties can be
| |
| − | solved by the Wiener-Hopf method. The first work on this problem was by [[Evans_Davies_1968a|Evans and Davies 1968]]
| |
| − | but they did not actually develop the method sufficiently to be able to calculate the solution.
| |
| − | The explicit solution was not found until the work of ...
| |
| − | | |
| − | ====[[Eigenfunction Matching Method]]====
| |
| − | | |
| − | The eigenfunction matching solution was developed by [[Fox_Squire_1994a|Fox and Squire 1994]].
| |
| − | Essentially the solution is expanded on either side of the crack.
| |
| − | | |
| − | ====[[Residue Calculus]]====
| |
| | | | |
| | = Three Dimensional Problem = | | = Three Dimensional Problem = |
| Line 193: |
Line 80: |
| | | | |
| | <math>D\nabla ^4 w + \rho _i h w = p</math> | | <math>D\nabla ^4 w + \rho _i h w = p</math> |
| − |
| |
| − | ==The Equation of Motion for the Ice Floe==
| |
| − |
| |
| − |
| |
| − |
| |
| − | Ice floes range in size from much smaller to much larger than the dominant
| |
| − |
| |
| − | wavelength of the ocean waves. However there are two reasons why solutions
| |
| − |
| |
| − | for ice floes of intermediate size (a size similar to the wavelength) are
| |
| − |
| |
| − | the most important. The first is that at these intermediate sizes ice floes
| |
| − |
| |
| − | scatter significant wave energy. The second is that, since it is wave
| |
| − |
| |
| − | induced flexure which determines the size of ice floes in the MIZ, ice floes
| |
| − |
| |
| − | tend to form most often at this intermediate length.
| |
| − |
| |
| − |
| |
| − |
| |
| − | The theory for an ice floe of intermediate size which is developed in this
| |
| − |
| |
| − | paper obviously also applies to small or large floes. However, if the
| |
| − |
| |
| − | solution for a small or large floe is required then the appropriate simpler
| |
| − |
| |
| − | theory should be used. Small ice floes (ice floes much small than the
| |
| − |
| |
| − | wavelength) should be modelled as rigid \citep{Masson_Le,Massondrift}. Large
| |
| − |
| |
| − | ice floes (ice floes much larger than the wavelength) should be modelled as
| |
| − |
| |
| − | infinite and flexible \citep{FoxandSquire}. In the intermediate region,
| |
| − |
| |
| − | where the size of the wavelength is similar to the size of the ice floe, the
| |
| − |
| |
| − | ice floe must be modelled as finite and flexible.
| |
| − |
| |
| − |
| |
| − |
| |
| − | We model the ice floe as a thin plate of constant thickness and shallow
| |
| − |
| |
| − | draft following \citet{Wadhams1986} and \citet{Squire_Review}. The thin
| |
| − |
| |
| − | plate equation \citep{Hildebrand65} gives the following equation of motion
| |
| − |
| |
| − | for the ice floe
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | D\nabla ^{4}W+\rho _{i}h\frac{\partial ^{2}W}{\partial t^{2}}=p,
| |
| − |
| |
| − | \label{plate}
| |
| − |
| |
| − | </math></center>
| |
| − | where <math>W</math> is the floe displacement, <math>\rho _{i}</math> is the floe density, <math>h</math> is
| |
| − |
| |
| − | the floe thickness, <math>p</math> is the pressure, and <math>D</math> is the modulus of rigidity
| |
| − |
| |
| − | of the floe (<math>D=Eh^{3}/12(1-\nu ^{2})</math> where <math>E</math> is the Young's modulus and <math>
| |
| − | \nu <math> is Poisson's ratio). Visco-elastic effects can be included by making </math>
| |
| − | D </math> have some imaginary (damping)\ component but this will not be done here
| |
| − |
| |
| − | to keep the presented results simpler. We assume that the plate is in
| |
| − |
| |
| − | contact with the water at all times so that the water surface displacement
| |
| − |
| |
| − | is also <math>W.</math> Equation (\ref{plate}) is subject to the free edge boundary
| |
| − |
| |
| − | conditions for a thin plate
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | \dfrac{\partial ^{2}W}{\partial n^{2}}+\nu \dfrac{\partial ^{2}W}{\partial
| |
| − |
| |
| − | s^{2}}=0,\;\;\;\text{\textrm{and}}\mathrm{\;\;\;}\dfrac{\partial ^{3}W}{
| |
| − | \partial n^{3}}+\left( 2-\nu \right) \dfrac{\partial ^{3}W}{\partial
| |
| − |
| |
| − | n\partial s^{2}}=0, \label{boundaryplate}
| |
| − |
| |
| − | </math></center>
| |
| − | \citep{Hildebrand65} where <math>n</math> and <math>s</math> denote the normal and tangential
| |
| − |
| |
| − | directions respectively.
| |
| − |
| |
| − |
| |
| − |
| |
| − | The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
| |
| − |
| |
| − | water surface,
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | p=-\rho \frac{\partial \Phi }{\partial t}-\rho gW \label{pressure}
| |
| − |
| |
| − | </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.
| |
| − |
| |
| − |
| |
| − |
| |
| − | We now introduce non-dimensional variables. We non-dimensionalise the length
| |
| − |
| |
| − | 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
| |
| − |
| |
| − | the mass variables with respect to <math>\rho a^{3}</math>. The non-dimensional
| |
| − |
| |
| − | variables, denoted by an overbar, are
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | \bar{x}=\frac{x}{a},\;\;\bar{y}=\frac{y}{a},\;\;\bar{z}=\frac{z}{a},\;\;\bar{
| |
| − | W}=\frac{W}{a},\;\;\bar{t}=t\sqrt{\frac{g}{a}},\;\;\text{and}\;\;\bar{\Phi}=
| |
| − | \frac{\Phi }{a\sqrt{ag}}.
| |
| − |
| |
| − | </math></center>
| |
| − | In the non-dimensional variables equations (\ref{plate}) and (\ref{pressure}
| |
| − | ) become
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | \beta \nabla ^{4}\bar{W}+\gamma \frac{\partial ^{2}\bar{W}}{\partial \bar{t}
| |
| − | ^{2}}=\frac{\partial \bar{\Phi}}{\partial \bar{t}}-\bar{W}, \label{n-d_ice}
| |
| − |
| |
| − | </math></center>
| |
| − | where
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | \beta =\frac{D}{g\rho a^{4}}\;\;\text{and\ \ }\gamma =\frac{\rho _{i}h}{\rho
| |
| − |
| |
| − | a}.
| |
| − |
| |
| − | </math></center>
| |
| − | We shall refer to <math>\beta </math> and <math>\gamma </math> as the stiffness and mass
| |
| − |
| |
| − | respectively.
| |
| − |
| |
| − |
| |
| − |
| |
| − | We will determine the response of the ice floe to wave forcing of a single
| |
| − |
| |
| − | frequency (the response for more complex wave forcing can be found by
| |
| − |
| |
| − | superposition of the single frequency solutions). Since the equations of
| |
| − |
| |
| − | motion are linear the displacement and potential must have the same single
| |
| − |
| |
| − | frequency dependence. Therefore they can be expressed as the real part of a
| |
| − |
| |
| − | 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},
| |
| − | \bar{y},\bar{t})=\func{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})=
| |
| − | \func{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 (\ref{n-d_ice}) is
| |
| − |
| |
| − | <center><math>
| |
| − |
| |
| − | \beta \nabla ^{4}w+\alpha \gamma w=\sqrt{\alpha }\phi -w. \label{plate2}
| |
| − |
| |
| − | </math></center>
| |
| − | From now on we will drop the overbar and assume all variables are
| |
| − |
| |
| − | non-dimensional.
| |
| − |
| |
| − |
| |
| − |
| |
| − | ==Equations of Motion for the Water==
| |
| − |
| |
| − |
| |
| − |
| |
| − | We require the equation of motion for the water to solve equation (\ref
| |
| − | {plate2}). We begin with the non-dimensional equations of potential theory
| |
| − |
| |
| − | which describe linear surface gravity waves
| |
| − |
| |
| − | <center><math> \label{bvp}
| |
| − |
| |
| − | \left.
| |
| − |
| |
| − | \begin{array}{rr}
| |
| − |
| |
| − | \nabla ^{2}\phi =0, & -\infty <z<0, \\
| |
| − |
| |
| − | {\dfrac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
| |
| − |
| |
| − | {\dfrac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
| |
| − | \mathbf{x}\in \Delta , \\
| |
| − |
| |
| − | {\dfrac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
| |
| − | x}\notin \Delta ,
| |
| − | \end{array}
| |
| − | \right\} \label{bvp_nond}
| |
| − |
| |
| − | </math></center>
| |
| − | (\citet{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
| |
| − |
| |
| − | 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 \ref{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.}}
| |
| − |
| |
| − | \label{vibration}
| |
| − |
| |
| − | \end{figure}
| |
| − |
| |
| − |
| |
| − |
| |
| − | The boundary value problem (\ref{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, \label{summerfield}
| |
| − |
| |
| − | </math></center>
| |
| − | \citep{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}, \label{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 (\ref{bvp}) and (\ref{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}}. \label{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>
| |
| − | \citep{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 \citep{abr_ste}. A solution for
| |
| − |
| |
| − | 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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Introduction
The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
physical structures such as a floating break water, an ice floe or a VLFS). The equations of motion were formulated
more than 100 years ago and a discussion of the problem appears in Stoker 1957. The problem can
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.
Two Dimensional Problem
Equations of Motion
When considering a two dimensional problem, the [math]\displaystyle{ 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.
For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given
by the following
[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p }[/math]
where [math]\displaystyle{ D }[/math] is the flexural rigidity, [math]\displaystyle{ \rho_i }[/math] is the density of the beam,
[math]\displaystyle{ h }[/math] is the thickness of the beam (assumed constant), [math]\displaystyle{ p }[/math] is the pressure
and [math]\displaystyle{ \eta }[/math] is the beam vertical displacement.
The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).
[math]\displaystyle{ \frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0 }[/math]
at the edges of the plate.
The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
pressure at the surface), i.e.
[math]\displaystyle{ p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t} }[/math]
where [math]\displaystyle{ \rho }[/math] is the water density and [math]\displaystyle{ g }[/math] is gravity, and [math]\displaystyle{ \phi }[/math]
is the velocity potential. The velocity potential is governed by Laplace's equation through out
the fluid domain subject to the free surface condition and the condition of no flow through the
bottom surface. If we denote the beam-covered (or possible multiple beams covered) region of the fluid by [math]\displaystyle{ P }[/math] and the free surface by [math]\displaystyle{ F }[/math] the equations of motion for the
Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] for water of
Finite Depth are the following. At the surface
we have the dynamic condition
[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
i\omega \rho \phi, \, z=0, \, x\in P }[/math]
[math]\displaystyle{ 0=
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F }[/math]
and the kinematic condition
[math]\displaystyle{ \frac{\partial\phi}{\partial z} = i\omega\eta }[/math]
The equation within the fluid is governed by Laplace's Equation
[math]\displaystyle{ \nabla^2\phi =0 }[/math]
and we have the no-flow condition through the bottom boundary
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-h }[/math]
(so we have a fluid of constant depth with the bottom surface at [math]\displaystyle{ z=-h }[/math] and the
free surface or plate covered surface are at [math]\displaystyle{ z=0 }[/math]).
[math]\displaystyle{ g }[/math] is the acceleration due to gravity, [math]\displaystyle{ \rho_i }[/math] and [math]\displaystyle{ \rho }[/math]
are the densities of the plate and the water respectively, [math]\displaystyle{ h }[/math] and [math]\displaystyle{ D }[/math]
the thickness and flexural rigidity of the plate.
Solution Method
There are many different methods to solve the corresponding equations ranging from highly analytic such
as the Wiener-Hopf to very numerical based on Eigenfunction Matching Method which are
applicable and have advantages in different situations.
Three Dimensional Problem
Equations of Motion
For a classical thin plate, the equation of motion is given by
[math]\displaystyle{ D\nabla ^4 w + \rho _i h w = p }[/math]
