Green Function Methods for Floating Elastic Plates

Introduction

The problem of a two-dimensional Floating Elastic Plate was solved using a Free-Surface Green Function by Newman 1994 and Meylan and Squire 1994. We describe here both methods (which are closely related).

The closest solution to the one presented here was derived by Hermans 2004, based on an earlier solution for a single plate Hermans 2003. This solution was for a set of finite elastic plates of arbitrary properties. That problem differed from the one presented here, only by requiring that the semi-infinite regions are open water. The solution method presented in Hermans 2004 was quite different from the one presented here, and it was based on using the free-surface Green function.

Equations of Motion

We begin with the equations of motion in non-dimensional form for a single Floating Elastic Plate which occupies the region [math]\displaystyle{ -b\leq x\leq b }[/math]. The full derivation of these equation is presented in Eigenfunction Matching Method for Floating Elastic Plates. We assume that the plate is infinite in the [math]\displaystyle{ y }[/math] direction, but we allow the wave to be incident at an angle which we do by introducing a wavenumber [math]\displaystyle{ k_y }[/math]. These means that the total potential is given by

[math]\displaystyle{ \Phi(x,y,z,y) = \Re\left(\phi(x,z)e^{i\omega t}e^{i k_y y}\right). }[/math]

The free-surface is at [math]\displaystyle{ z=0 }[/math] and the sea floor is at [math]\displaystyle{ z=-h }[/math]

[math]\displaystyle{ \begin{matrix} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} - k_y^2\right) \phi = 0 \;\;\;\; \mbox{ for } -h \lt z \leq 0, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial \phi}{\partial z} = 0 \;\;\;\; \mbox{ at } z = - h, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \left( \beta \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 - \gamma\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\; \mbox{ at } z = 0, \;\;\; -b \leq x \leq b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\; \mbox{ at } z = 0, \;\;\; x\lt -b \,\,\mathrm{or}\,\, b\lt x, \end{matrix} }[/math]

where [math]\displaystyle{ \alpha = \omega^2 }[/math] and

[math]\displaystyle{ \begin{matrix} \left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix}(17) \left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b. \end{matrix} }[/math]