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).
A related paper was given by Hermans 2003and we extended to
multiple plates in Hermans 2004.
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]
