View source for Template:Finite floating body on the surface frequency domain

We consider the problem of small-amplitude waves which are incident on finite floating body occupying water surface for <math>-L<x<L</math>. 
The submergence of the body is considered negligible. 
We assume that the problem is invariant in the <math>y</math> direction.
{{general dock type body equations}}
where <math>\alpha = \omega^2</math>. The equation under the body consists of
the kinematic condition
<center>
<math>
\mathrm{i}\omega w = \partial_z \phi,\,\,\, z=0,\,\,-L\leq x\leq L
</math>
</center>
plus the kinematic condition. The body motion is expanded using the modes for
heave and pitch. 
Using the expression <math>\partial_n \phi =\partial_t w</math>, we can form
<center>
<math>
\frac{\partial \phi}{\partial z} = i\omega \sum_{n=0,1} \xi_n X_n(x)
</math>
</center>
where <math>\xi_n \,</math> are coefficients to be evaluated.
The functions <math>X_n(x)</math> are given by
{{rigid modes for an elastic plate}}
Note that this numbering is non-standard for a floating body and comes
from [[Eigenfunctions for a Uniform Free Beam]].