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]\displaystyle{ -L\lt x\lt L }[/math]. The submergence of the body is considered negligible. We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction.
where [math]\displaystyle{ \alpha = \omega^2 }[/math]. The equation under the body consists of the kinematic condition
[math]\displaystyle{ \mathrm{i}\omega w = \partial_z \phi,\,\,\, z=0,\,\,-L\leq x\leq L }[/math]
plus the kinematic condition. The body motion is expanded using the modes for heave and pitch. Using the expression [math]\displaystyle{ \partial_n \phi =\partial_t w }[/math], we can form
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = i\omega \sum_{n=0,1} \xi_n X_n(x) }[/math]
where [math]\displaystyle{ \xi_n \, }[/math] are coefficients to be evaluated. The functions [math]\displaystyle{ X_n(x) }[/math] are given by
[math]\displaystyle{ X_0 = \frac{1}{\sqrt{2L}} }[/math]
and
[math]\displaystyle{ X_1 = \sqrt{\frac{3}{2L^3}} x }[/math]
Note that this numbering is non-standard for a floating body and comes from Eigenfunctions for a Uniform Free Beam.