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).
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]
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
and they are referred to as the added mass and damping coefficients (see Added-Mass, Damping Coefficients And Exciting Forces
for the equivalent definition for a rigid body).
respectively.
This equation is solved by truncating the number of modes.
Solution for the Radiation and Diffracted Potential
We use the Free-Surface Green Function for two-dimensional waves, with singularity at
the water surface 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^{(d)}(x) = \phi^{i}(x) + \int_{-L}^{L}G(x,\xi)
\alpha\phi^{(d)}(\xi) d \xi
}[/math]
We can calculate the Reflection and Transmission coefficients as follows
A diagram depicting the area [math]\displaystyle{ \mathcal{U} }[/math] which is bounded by the rectangle [math]\displaystyle{ \mathcal{S} }[/math].] {A diagram depicting the area [math]\displaystyle{ \mathcal{U} }[/math] which is bounded by the rectangle [math]\displaystyle{ \mathcal{S} }[/math]. The rectangle [math]\displaystyle{ \mathcal{S} }[/math] is bounded by [math]\displaystyle{ -h\leq z \leq0 }[/math] and [math]\displaystyle{ -\infty\leq x \leq \infty }[/math]
Applying Green's theorem to [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \phi^{i} }[/math] gives
and using a wave incident from the right we obtain
[math]\displaystyle{
1 + T = k_0 \frac{\int_{-L}^{L} e^{-k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)dx }
{\int_{-h}^{0} \left(\phi_0(z)\right)^2 dz}.
== Matlab Code ==
A program to calculate the coefficients for the semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/boundary_element/elastic_plate_modes.m elastic_plate_modes.m]
=== Additional code ===
This program requires
* {{green function surface code}}
* {{free surface dispersion equation code}}
== Alternative Solution Method using Green Function for the Plate ==
We can also solve the equation by a closely related method which was given in
[[Meylan and Squire 1994]].
We can transform the equations to
\lt center\gt \lt math\gt
\phi(x) = \phi^{\rm In}(x) + \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi(\xi) - \partial_z\phi(\xi)
\right)d \xi
}[/math]
