Two-Dimensional Floating Elastic Plate

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.

Energy Balance

An energy balence relation is derived in Evans and Davies 1968 which is simply a condition that the incident energy is equal to the sum of the radiated energy including both the energy in the water and the energy in the plate. If the properties of the first and last semi-infinite plates were identical, then this would be the familiar requirement that

[math]\displaystyle{ |T_{\Lambda}(0)|^2 + |R_{1}(0)|^2 = |I|^2. }[/math]

However, when the first and last plates have different properties, then the energy balance condition becomes the following

[math]\displaystyle{ D|T_{\Lambda}(0)|^2 + |R_{1}(0)|^2 = |I|^2, }[/math]

where [math]\displaystyle{ D }[/math] is found by applying Green's theorem to [math]\displaystyle{ \phi }[/math] and its conjugate, Evans and Davies 1968 and is given by

[math]\displaystyle{ \begin{matrix} D & = & { \frac{\kappa_{\Lambda}(0)k_1(0)\cosh^2{(k_1(0)h)}}{\kappa_{1}(0)k_\Lambda(0)\cosh^2{(k_\Lambda(0)h)}} \times }\\ & & { \frac{\left(\frac{\beta_\Lambda}{\alpha}4k_\Lambda(0)^3(\kappa_{\Lambda}(0)^2 +k_y^2)\sinh^2{(k_\Lambda(0) h)} + \frac{1}{2}{\sinh{(2k_\Lambda(0)h)}}+k_\Lambda(0)h\right)} {\left(\frac{\beta_1}{\alpha}4k_1(0)^3(\kappa_{1}(0)^2 +k_y^2)\sinh^2{(k_1(0)h)} + \frac{1}{2}{\sinh{(2k_1(0)h)}}+k_1(0)h\right)}. } \end{matrix} }[/math]

The energy balance condition is useful to help check that the solution is not incorrect (it does not of course guarantee the solution is correct). The energy balance condition is surprisingly well satisfied by our solutions, for example with [math]\displaystyle{ M=20 }[/math] we can easily get ten decimal places.

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 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]\displaystyle{ 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]\displaystyle{ |R| }[/math] could be calculated by simply using the correct value for [math]\displaystyle{ 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

Incomplete mode-matching approaches were then taken (e.g. Hendrickson and Webb 1963, Wadhams 1973,Squire 1978, Squire 1984, Wadhams 1986), before 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. The eigenfunction matching solution was developed by Fox and Squire 1994 for a single semi-infinite plate and extended to plates of different properties by Barrett and Squire 1996.

Other authors have returned to the mode-matching approach of Fox and Squire 1990. Sahoo et al. 2001 defined an inner product enabling the solution to be found by using [math]\displaystyle{ N }[/math] eigenfunctions and inverting an [math]\displaystyle{ N\times N }[/math] matrix, while Linton and Chung 2003 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. 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).

Essentially the solution is expanded on either side of the crack. The theory is described in Eigenfunction Matching Method for Floating Elastic Plates

Residue Calculus

The solution using Residue Calculus was developed by Linton and Chung 200?

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).