View source for Green Function Methods for Floating Elastic Plates

{{complete pages}}

== Introduction ==

The problem of a two-dimensional [[:Category:Floating Elastic Plate|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 this was 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]].

The simpler problem of dock is treated in [[Green Function Method for Finite Dock]]

== Equations for a Finite Plate in Frequency Domain ==
{{finite plate frequency domain}}

== Transformation using [[Eigenfunctions for a Uniform Free Beam]] ==

{{equations for a eigenfunction of a free beam}}

This solution is discussed further in [[Eigenfunctions for a Free Beam]]. <br /><br />
Using the expression <math>\partial_n \phi =\partial_t w</math>, we can form
<center>
<math>
\frac{\partial \phi}{\partial z} = -\mathrm{i}\omega \sum_{n=0}^{\infty} \zeta_n X_n
</math>
</center>
where <math>\zeta_n \,</math> are coefficients to be evaluated.

== Equation in Terms of the Modes of the Plate ==

Under these assumptions, the equations become
<center><math>
\Delta\phi   =0,\,\,-h<z<0,
</math></center>
<center><math>
\partial_{z}\phi   =0,\,\,z=-h,
</math></center>
<center><math>
\alpha\phi  =\partial_{z}\phi,\,\,x\notin(-L,L),\ \ z=0,
</math></center>
<center><math>
i\omega\sum_{n=0}^{\infty}\zeta_{n}X_{n}    =\partial_{z}\phi,\,\,x\in
(-L,L),\,\, z=0,
</math></center>
<center><math>
\sum_{n=0}^{\infty}\zeta_{n}\left(  1+\beta\lambda_{n}^{4}\right)
X_{n}-\alpha\gamma\sum_{n=0}^{\infty}\zeta_{n}X_{n}    = -i\omega
\phi,\,\,x\in(-L,L),\,\, z=0.
</math></center>
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.
<center><math>
\phi=\phi^{\mathrm{D}}+\phi^{\mathrm{R}} ,\,
</math></center>
We begin with the diffraction potential <math>\phi^{\mathrm{D}}</math> which
satisfies the following equations
{{diffraction potential equations for a dock}}
{{radiation condition for diffracted potential}}

As the plate is floating on the surface, we can denote it as follows:
<center><math>
\phi^{\rm I}|_{z=0}  =  e^{-k_0 x} \,
</math></center>
where we have set the amplitude to be unity. 

We now consider the scattered potentials <math>\phi^{\mathrm{S}}</math>.  The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:
<center><math>
\phi^{\mathrm{D}}=\phi^{\mathrm{I}}+\phi^{\mathrm{S}} \,
</math></center>
from this, we can construct the following conditions:
<center><math>
\Delta\phi^{\mathrm{S}}    =0,\,\,-h<z<0,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{S}}    =0,\,\,z=-h,
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{S}}    =\alpha\phi^{\mathrm{S}},\,\,x\notin(-L,L),\, \,
z=0
</math></center>
<center><math>
\partial_{z}\phi^{\mathrm{S}}    = -\partial_{z}\phi^{\mathrm{I}},\,\,x\in(-L,L),\,\,z=0.
</math></center> <br /><br />

We now consider the radiation potentials <math>\phi^{\mathrm{R}}</math>.  We can express the radiation potential as:
<center><math>
\phi^{\mathrm{R}}=\sum_{n=0}^{\infty}\zeta_n \phi_n^{\mathrm{R}}
</math></center>
which satisfy the following equations

{{radiation potential equations for a dock like structure}}

Therefore we find the potential as
<center><math>
\phi=\phi^{\mathrm{D}} +\sum_{n=0}^{\infty}\zeta_{n}\phi_n^{\mathrm{R}},
</math></center>
so that
<center><math>
\sum_{n=0}^{\infty}\left( 1+\beta\lambda_{n}^{4} - \alpha\gamma\right)
\zeta_{n}X_{n}=-i\omega \left( \phi^{\mathrm{D}}+\sum_{n=0}^{\infty}\zeta_{n}\phi_n^{\mathrm{R}} \right).
</math></center>
If we multiply by <math>X_m</math> and take an inner product over the plate we obtain
<center><math>
\left(  1+\beta\lambda_{n}^{4} - \alpha\gamma\right)  \zeta_{n}=-i\omega
\int_{-L}^{L}\phi^{\mathrm{D}} X_{n}\mathrm{d}x +
\sum_{m=0}^{\infty}\left(\omega^2 a_{mn}(\omega) - i\omega b_{mn}(\omega)\right)
\zeta_{m},
</math></center>
where the functions <math>a_{mn}(\omega)</math> and <math>b_{mn}(\omega)</math> are given by
<center><math>
\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,
</math></center>
and they are referred to as the added mass and damping coefficients (see [[Linear Wave-Body Interaction]]
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 ==

{{Green's function equations for the diffracted potential}}

and 

{{Green's function equations for the radiation potential for a dock like structure}}

== Reflection and Transmission Coefficients ==

{{derivation of reflection and transmission in two dimensions}}

== Matlab Code ==

{{elastic plate modes code}}

== Alternative Solution Method using Green Functions for a Uniform 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
<center><math>
\phi(x) = \phi^{\rm I}(x) + \int_{-L}^{L}G(x,\xi)
\left(
\alpha\phi(\xi) - \partial_z\phi(\xi)
\right)\mathrm{d} \xi
</math></center>

Expanding as before
<center>
<math>
\partial_z \phi = i\omega \sum \xi_n X_n
</math>
</center>
we obtain
<center><math>
-i\omega \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)\xi_n X_n
</math>
</center>
This leads to the following equation 
<center>
<math>
\partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1} \phi(\xi)\mathrm{d}\xi
</math>
</center>
or 
<center>
<math>
\partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} g(x,\xi) \phi(\xi)\mathrm{d}\xi
</math>
</center>
where 
<center>
<math>
g(x,\xi) = \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}
</math>
</center>
which is the Green function for the plate.

[[Category:Floating Elastic Plate]]