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.

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

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