Two-Dimensional Floating Elastic Plate

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Equations of Motion

There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler Beam theory (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, the equation of motion is given by the following

[math] \partial_x^2\left(\beta(x)\partial_x^2 \zeta\right) + \gamma(x) \partial_t^2 \zeta = p [/math]

where [math]\beta(x)[/math] is the non dimensionalised flexural rigidity, and [math]\gamma [/math] is non-dimensionalised linear mass density function. Note that this equations simplifies if the plate has constant properties (and that [math]h[/math] is the thickness of the plate, [math] p[/math] is the pressure and [math]\zeta[/math] is the plate vertical displacement) .

The edges of the plate can satisfy a range of boundary conditions. The natural boundary condition (i.e. free-edge boundary conditions).

[math] \partial_x^2 \zeta = 0, \,\,\partial_x^3 \zeta = 0 [/math]

at the edges of the plate.

The problem is subject to the initial conditions

[math] \zeta(x,0)=f(x) \,\! [/math]
[math] \partial_t \zeta(x,0)=g(x) [/math]

We derive here the equations for a plate on a fluid, ignoring boundary conditions at the plate edge and assuming the plate occupies the entire fluid region.

We begin with the linear equations for a fluid. The kinematic condition is the same

[math] \frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; [/math]

but the dynamic condition needs to be modified to include the effect of the the plate

[math] \rho g\zeta + \rho \frac{\partial\Phi}{\partial t} = D \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} , \ z=0; [/math]

We also have Laplace's equation

[math] \Delta \Phi = 0,\,\,-h\ltz\lt0 [/math]

and the usual non-flow condition at the bottom surface

[math] \partial_z \Phi = 0,\,\,z=-h, [/math]

where [math]\zeta[/math] is the surface displacement, [math]\Phi[/math] is the velocity potential, and [math]\rho[/math] is the fluid density.

Frequency Domain Equations

If we make the assumption of Frequency Domain Problem that everything is proportional to [math]\exp (-\mathrm{i}\omega t)\,[/math] the equations become

[math] \begin{align} -\mathrm{i}\omega\zeta &= \partial_z\phi , &z=0 \\ \rho g\zeta - \mathrm{i}\omega\rho \phi &= D \partial_x^4 \eta -\omega^2 \rho_i h \zeta, &z=0 \\ \Delta \phi &= 0, &-h\ltz\lt0 \\ \partial_z \phi &= 0, &z=-h, \end{align} [/math]

where [math]\zeta[/math] is the surface displacement and [math]\phi[/math] is the velocity potential in the frequency domain.

These equations can be simplified by defining [math]\alpha = \omega^2/g[/math], [math]\beta = D/\rho g[/math] and [math]\gamma = \rho_i h/\rho[/math] to obtain

[math] \begin{align} \Delta \phi &= 0, &-h \lt z \leq 0 \\ \partial_z \phi &= 0, &z = - h \\ \beta \partial_x^4 \zeta + \left( 1 - \gamma\alpha \right) \zeta &= -\mathrm{i} \sqrt{\alpha}\phi, &z = 0 \\ -\mathrm{i}\omega\zeta &= \partial_z\phi , &z=0 . \end{align} [/math]

Computational Methods

This wiki contains a number of methods to solve for a floating elastic plate in the frequency domain.

Solution Methods

There are many different methods to solve the corresponding equations ranging from highly analytic such as the Wiener-Hopf to very numerical based on the Eigenfunction Matching Method which are applicable and have advantages in different situations. We describe here some of the solutions which have been developed, grouped by problem

Two Semi-Infinite Plates of Identical Properties

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 which is described in Two Semi-Infinite Elastic Plates of Identical Properties

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 and Davies 1968 but they did not actually develop the method sufficiently to be able to calculate the solution. At the time the infinite products that formed a large part of the solution proved to be too difficult to compute and so they were not able to present any results, except for certain limiting cases (such as the shallow water limit). This was because at the time the infinite products that formed a large part of the solution proved to be too difficult to compute and so they were not able to present general results. Gol'dshtein and Marchenko 1989 presented a later Wiener-Hopf solution for infinite depth, but again only certain limiting situations were discussed. However, in the late 1990's and early [math]21^{st}[/math] century improvements in computing power enabled other authors to compute results using the Wiener-Hopf solution of Evans and Davies 1968. Balmforth and Craster 1999 turned the required infinite products into integrals which were evaluated by quadrature, while Chung and Fox 2002 showed that the products themselves could be evaluated directly with relatively little effort. Ironically Tkacheva 2001 finally showed that if the inertia term in the thin plate equation was neglected for normally incident waves (as it can for most wavelengths), then [math]|R|[/math] could be calculated by simply using the correct value for [math]k_0[/math], the wavenumber in the ice, in the formula for Mass Loading Model of Ice given by Keller and Weitz 1953. The Wiener-Hopf solution is described in Wiener-Hopf Elastic Plate Solution

Eigenfunction Matching Method

Essentially the solution is expanded on either side of the crack using separation of variables. The theory is described in Eigenfunction Matching Method for Floating Elastic Plates The eigenfunction matching method using incomplete mode-matching was used by Hendrickson and Webb 1963, Wadhams 1973, Squire 1978, Squire 1984, Wadhams 1986. Fox and Squire 1990 computed the solution for the full set of eigenfunctions using a conjugate gradient technique. Fox and Squire 1994 used the same method to complete further studies on this problem, investigating the strain in the ice, and also the effect of shore fast ice on an incoming directional wave spectrum of specified structure.

Other authors have extended mode-matching approach of Fox and Squire 1990. The eigenfunction matching solution was extended to plates of different properties by Barrett and Squire 1996. Sahoo et al. 2001 defined an inner product enabling the solution to be found by using [math]N[/math] eigenfunctions and inverting an [math]N\times N[/math] matrix, Earlier Chakrabarti 2000 had used an infinite depth mode-matching scheme to set up a singular integral equation, the equivalent problem to a residue calculus problem when the eigenvalue spectrum is continuous. This singular integral equation was solved by transforming it into a Hilbert problem (Roos 1969), a generalization of the Wiener-Hopf problem (although in practice both are solved in exactly the same way).

Residue Calculus

The solution using Residue Calculus was developed Linton and Chung 2003 who effectively showed that the equations Sahoo et al. 2001 had set up could be solved analytically using Residue Calculus. In the process they also demonstrated its equivalence to the Wiener-Hopf solution, and confirmed the formula of Tkacheva 2001.

Single Floating Plate

The problem of a single floating plate in two-dimensions was treated by Newman 1994, Meylan and Squire 1994 and Hermans 2003 using the Free-Surface Green Function, described in Green Function Methods for Floating Elastic Plates

Multiple Floating Plates

The most general problem consists of multiple floating plate. The methods which generalises to this are the Eigenfunction Matching Method for Floating Elastic Plates (Kohout et. al. 2006) and the Green Function Methods for Floating Elastic Plates (Hermans 2004).