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 2003 and 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}
\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]
Transformation Using the Green function
We use the Free-Surface Green Function for two-dimensional waves incident
at an angle which we denote by [math]\displaystyle{ G(x,\zeta) }[/math] since we are only
interested in its value at [math]\displaystyle{ z=0. }[/math] Using this we can transform the
system of equations to
[math]\displaystyle{
\phi(x) = \phi^{i}(x) + \int_{-b}^{b}G(x,\xi)
\left(
\alpha\phi(\xi) - \phi_z(\xi)
\right)d \xi
}[/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}
\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}
\left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0
\;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b.
\end{matrix} }[/math]
We will consider now the case where [math]\displaystyle{ k_y=0 }[/math], although the solutions presented
here can be generalised to the case when [math]\displaystyle{ k_y\neq 0 }[/math]. Under this assumption
the equations reduce to
[math]\displaystyle{
\phi(x) = \phi^{i}(x) + \int_{-b}^{b}G(x,\xi)
\left(
\alpha\phi(\xi) - \phi_z(\xi)
\right)d \xi
}[/math]
[math]\displaystyle{ \begin{matrix}
\left( \beta \frac{\partial^4}{\partial x^4}
- \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^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b,
\end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b.
\end{matrix} }[/math]
Transformation using the Eigenfuctions for the plate
We can find a the eigenfunction which satisfy
[math]\displaystyle{ \partial_x^4 w_n = \lambda_n^4 w_n }[/math]
plus the edge conditions.
[math]\displaystyle{ \begin{matrix}
\frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b,
\end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix}
\frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b.
\end{matrix} }[/math]This solution is discussed further in Eigenfunctions for a Beam.
Expanding
[math]\displaystyle{ \partial_x^4 w_n = \lambda_n^4 w_n }[/math]
