# Template:Finite floating body on the surface frequency domain

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We consider the problem of small-amplitude waves which are incident on finite floating body occupying water surface for $\displaystyle{ -L\lt x\lt L }$. The submergence of the body is considered negligible. We assume that the problem is invariant in the $\displaystyle{ y }$ direction.

$\displaystyle{ \Delta \phi = 0, \;\;\; -h \lt z \leq 0, }$
$\displaystyle{ \partial_z \phi = 0, \;\;\; z = - h, }$
$\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L }$

where $\displaystyle{ \alpha = \omega^2 }$. The equation under the body consists of the kinematic condition

$\displaystyle{ \mathrm{i}\omega w = \partial_z \phi,\,\,\, z=0,\,\,-L\leq x\leq L }$

plus the kinematic condition. The body motion is expanded using the modes for heave and pitch. Using the expression $\displaystyle{ \partial_n \phi =\partial_t w }$, we can form

$\displaystyle{ \frac{\partial \phi}{\partial z} = i\omega \sum_{n=0,1} \xi_n X_n(x) }$

where $\displaystyle{ \xi_n \, }$ are coefficients to be evaluated. The functions $\displaystyle{ X_n(x) }$ are given by

$\displaystyle{ X_0 = \frac{1}{\sqrt{2L}} }$

and

$\displaystyle{ X_1 = \sqrt{\frac{3}{2L^3}} x }$

Note that this numbering is non-standard for a floating body and comes from Eigenfunctions for a Uniform Free Beam.