View source for Green Function Method for a Floating Body on the Surface

{{complete pages}}

== Introduction ==

The problem of a two-dimensional floating body which has negligible submergence
is solved using a green function. 
The problem of a dock is solved in [[Green Function Method for a Finite Dock]]
and for a floating elastic plate is solved 
in [[Green Function Methods for Floating Elastic Plates]]

== Equations for a Finite Plate in Frequency Domain ==

{{finite floating body on the surface frequency domain}}



== Equation in Terms of the Modes ==

The equations are
{{general dock type body equations}}
<center><math>
\mathrm{i}\omega\sum_{n=0,1}\zeta_{n}X_{n}    =\partial_{z}\phi,\,\,x\in
(-L,L),\,\, z=0,
</math></center>
<center><math>
\left(- \alpha\gamma + 1\right)  \zeta_{n}=-i\omega
\int_{-L}^{L}\phi X_{n}\mathrm{d}x, \,\,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}}

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=0,1}\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>
\left( - \alpha\gamma + 1\right)  \zeta_{n}=-i\omega
\int_{-L}^{L}\phi^{\mathrm{D}} X_{n}\mathrm{d}x +
\sum_{m=0,1}\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 [[Added-Mass, Damping Coefficients And Exciting Forces]])
respectively.

Note that for this simple example the added mass and damping matrices are diagonal.

== 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 ==

{{rigid plate modes code}}