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