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.
We present here the solution for a floating elastic plate using dry modes.
We begin with the equations. The solution can also be found using
Eigenfunction Matching for a Finite Floating Elastic Plate using Symmetry.
Equations for a Finite Plate in Frequency Domain
We consider the problem of small-amplitude waves which are incident on finite floating elastic
plate occupying water surface for [math]\displaystyle{ -L\lt x\lt L }[/math].
These equations are derived in Floating Elastic Plate
The submergence of the plate is considered negligible.
We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction.
We also assume that the plate edges are free to move at
each boundary, although other boundary conditions could easily be considered using
the methods of solution presented here. We begin with the Frequency Domain Problem for a semi-infinite
Floating Elastic Plates
in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate).
We also assume that the waves are normally incident (incidence at an angle will be discussed later).
[math]\displaystyle{
\Delta \phi = 0, \;\;\; -h \lt z \leq 0,
}[/math]
[math]\displaystyle{
\partial_z \phi = 0, \;\;\; z = - h,
}[/math]
[math]\displaystyle{
\partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L
}[/math]
[math]\displaystyle{
\partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\}
- \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;
z = 0, \;\;\; -L \leq x \leq L,
}[/math]
where [math]\displaystyle{ \alpha = \omega^2 }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math]
are the stiffness and mass constant for the plate respectively. The free edge conditions
at the edge of the plate imply
[math]\displaystyle{
\partial_x^3 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L,
}[/math]
[math]\displaystyle{
\partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L,
}[/math]
Expansion in Modes
We expand the plate motion in the free (dry) modes which satisfy
[math]\displaystyle{
\partial_{x}^{4}w_{n}=\lambda_{n}^4w_{n},
}[/math]
and the free-edge conditions
[math]\displaystyle{
\lim_{{x}\downarrow-{b}}\partial_{{x}}^{2}w_{n}=\lim_{{x}\uparrow{b}}
\partial_{{x}}^{2}w_{n}=\lim_{{x}\downarrow-{b}}\partial_{{x}}^{3}w_{n}
=\lim_{{x}\uparrow{b}}\partial_{{x}}^{3}w_{n}=0.
}[/math]
We further assume that the modes are normalised so that
[math]\displaystyle{
\int_{-b}^{b}w_{m}(x)w_{n}(x)dx=\delta_{mn},
}[/math]
where [math]\displaystyle{ \delta_{mn} }[/math] is the Kronecker symbol. The modes can be
calculated straightforwardly.
We use the kinematic equation
[math]\displaystyle{
i\omega w = \partial_z \phi
}[/math]
to expand the solution in the free (dry) modes of a plate.
Equation in Terms of the Modes of the Plate
Under these assumptions, the equations become
[math]\displaystyle{ \begin{matrix}
\Delta\phi & =0,\,\,-h\lt z\lt 0,\\
\partial_{z}\phi & =0,\,\,z=-h,\\
\omega^{2}\phi & =\partial_{z}\phi,\,\,x\notin(-b,b),\ \ z=0,\\
i\omega\sum_{n=0}^{\infty}\alpha_{n}w_{n} & =\partial_{z}\phi,\,\,x\in
(-b,b),\,\, z=0,\\
\sum_{n=0}^{\infty}\alpha_{n}\left( 1+\beta\lambda_{n}^{4}\right)
w_{n}-\omega^{2}\gamma\sum_{n=0}^{\infty}\alpha_{n}w_{n} & = -i\omega
\phi,\,\,x\in(-b,b),\,\, z=0.
\end{matrix} }[/math]
We solve for the potential (and displacement) as the sum of
the diffracted and radiation potentials in the standard way,
as for a rigid body.
We begin with the diffraction potential [math]\displaystyle{ \phi^{(d)} }[/math] which
satisfies the following equations
[math]\displaystyle{ \begin{matrix}
\Delta\phi^{(d)} & =0,\,\,-h\lt z\lt 0,\\
\partial_{z}\phi^{(d)} & =0,\,\,z=-h,\\
\partial_{z}\phi^{(d)} & =\omega^{2}\phi^{(d)},\,\,x\notin(-b,b),\,\,
z=0,\\
\partial_{z}\phi^{(d)} & =0,\,\,x\in(-b,b),\,\,z=0.
\end{matrix} }[/math]
Furthermore, [math]\displaystyle{ \phi^{(d)} }[/math] satisfies the radiation condition
[math]\displaystyle{
\frac{\partial}{\partial x} \left(\phi^{(d)}-\phi_{\kappa}^{\rm
In} \right) \pm ik\left( \phi^{(d)}-\phi^{\rm In}_{\kappa}\right)
= 0
,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty,
}[/math]
where
[math]\displaystyle{ k }[/math] is the wavenumber,
which is the positive real solution of the dispersion equation
[math]\displaystyle{
k\tanh(kh)=\omega^{2},
}[/math]
and [math]\displaystyle{ \phi^{\rm In}_{\kappa} }[/math] is the incident wave given by
[math]\displaystyle{
\phi^{\rm In}_\kappa = \frac{i\omega}{k\sinh kh} \cosh k(z+h) e^{i\kappa kx}
}[/math]
(which has unit amplitude in displacement) and
[math]\displaystyle{ \kappa }[/math] is either [math]\displaystyle{ 1 }[/math] for a wave travelling towards
negative infinity or [math]\displaystyle{ -1 }[/math] for a wave travelling towards positive
infinity
(we will need both these solutions).
We now consider the radiation potentials [math]\displaystyle{ \phi^{(n)} }[/math], which satisfy the
following equations
[math]\displaystyle{ \begin{matrix}
(11)
\Delta\phi^{(n)} & =0,\,\,-h\lt z\lt 0,\\
\partial_{z}\phi^{(n)} & =0,\,\,z=-h,\\
\partial_{z}\phi^{(n)} & =\omega^{2}\phi^{(n)},\,\,x\notin(-b,b),\, \,
z=0\\
(12)
\partial_{z}\phi^{(n)} & = i\omega w_{n},\,\,x\in(-b,b),\,\,z=0.
\end{matrix} }[/math]
The radiation condition for the radiation potential is
[math]\displaystyle{
(13)
\frac{\partial\phi^{(n)}}{\partial x}\pm ik\phi^{(n)}=0,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty.
}[/math]
The method used to solve these equations for deep water is described in
sturova06, and for finite depth in newman94 .
Therefore we find the potential as
[math]\displaystyle{
\phi=\phi^{(d)}_\kappa +\sum_{n=0}^{\infty}\alpha_{n,\kappa}\phi^{(n)},
}[/math]
so that
[math]\displaystyle{
\sum_{n=0}^{\infty}\left( 1+\beta\lambda_{n}^{4} - \omega^{2}\gamma\right)
\alpha_{n,\kappa}w_{n}=-i\omega\phi
^{(d)}_\kappa-i\omega\sum_{n=0}^{\infty}\alpha_{n,\kappa}\phi^{(n)}.
}[/math]
If we multiply by [math]\displaystyle{ w_m }[/math] and take an inner product over the plate we obtain
[math]\displaystyle{
(14)
\left( 1+\beta\lambda_{n}^{4}+\omega^{2}\gamma\right) \alpha_{n,\kappa
}=i\omega
\int_{-b}^{b}\phi^{(d)}_\kappa w_{n}dx +
\sum_{m=0}^{\infty}\left(-\omega^2 a_{mn}(\omega)+ i\omega b_{mn}(\omega)\right)
\alpha_{m,\kappa},
}[/math]
where the functions [math]\displaystyle{ a_{mn}(\omega) }[/math] and [math]\displaystyle{ b_{mn}(\omega) }[/math] are given by
[math]\displaystyle{
a_{mn}(\omega) + b_{mn}(\omega)/i\omega=\int_{-b}^{b}\phi^{(m)}w_{n}dx,
}[/math]
and they are referred to as the added mass and damping coefficients
respectively.
Equation \eqref{single_freq_numeric} is solved by truncating the
number of modes.
The expression for the displacement is
[math]\displaystyle{
(15)
\zeta_{\kappa}(x,\omega) = \left\{
\begin{matrix}[c]{ll}
\sum_{n=0}^{\infty}\alpha_{n,\kappa}(\omega)w_{n}(x),
\,\,x\in(-b,b), \\
-i\omega \left(\phi^{(d)}_\kappa(x,0) +
\sum_{n=0}^{\infty}\alpha_{n,\kappa}\phi^{(n)}(x,0) \right),\,\,x\notin(-b,b).
\end{matrix}
\right.
}[/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]
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{
\frac{\partial \phi}{\partial z} = \sum a_n w_n
}[/math]
we obtain
[math]\displaystyle{
\alpha \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)a_n w_n
}[/math]
This leads to the following equation
[math]\displaystyle{
\phi(x) = \frac{1}{\alpha} \int_{-b}^{b} \frac{w_n(x)w_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}\phi_z(\xi)d\xi
}[/math]
or
[math]\displaystyle{
\phi(x) = \frac{1}{\alpha} \int_{-b}^{b} g(x,\xi)\phi_z(\xi)d\xi
}[/math]
where
[math]\displaystyle{
g(x,\xi) = \frac{w_n(x)w_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}
}[/math]
