View source

== Equations of Motion ==

{{equations for a beam}}

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. 

{{linear elastic plate on water time domain}}

=== Frequency Domain Equations ===

{{frequency domain equations for a floating plate}}

== Computational Methods ==

This wiki contains a number of methods to solve for a floating elastic plate in the frequency domain. 
* [[Eigenfunction Matching for a Semi-Infinite Floating Elastic Plate]]
* [[Eigenfunction Matching for a Finite Floating Elastic Plate using Symmetry]]
* [[Eigenfunction Matching Method for Floating Elastic Plates]]
* [[Eigenfunction Matching for a Circular Floating Elastic Plate]]
* [[Green Function Methods for Floating Elastic Plates]]

== Solution Methods ==

There are many different methods to solve the corresponding equations ranging from highly analytic such
as the [[:Category:Wiener-Hopf|Wiener-Hopf]] to very numerical based on the [[:Category:Eigenfunction Matching Method|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 [[:Category:Residue Calculus|Residue Calculus]] solution which applies only when one of the semi-infinite regions
is water.

===[[:Category:Wiener-Hopf|Wiener-Hopf]]===

The solution to the problem of two semi-infinite plates with different properties can be
solved by the [[:Category:Wiener-Hopf|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 [[:Category:Wiener-Hopf|Wiener-Hopf]]
solution is described in [[Wiener-Hopf Elastic Plate Solution]]

===[[:Category:Eigenfunction Matching Method|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 [[:Category:Wiener-Hopf|Wiener-Hopf]] problem (although in practice both are solved in exactly the same way).

===[[:Category:Residue Calculus|Residue Calculus]]===

The solution using [[:Category:Residue Calculus|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 [[:Category:Residue Calculus|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]]).

[[Category:Floating Elastic Plate]]
[[Category:Complete Pages]]