Template:Finite plate frequency domain

We consider the problem of small-amplitude waves which are incident on finite floating elastic plate occupying water surface for [math]\displaystyle{ -L\lt x\lt L }[/math]. The submergence of the plate is considered negligible. We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction. We also assume that the plate edges are free to move at each boundary, although other boundary conditions could easily be considered using the methods of solution presented here. We begin with the Frequency Domain Problem for a semi-infinite Floating Elastic Plates in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate). We also assume that the waves are normally incident (incidence at an angle will be discussed later).

[math]\displaystyle{ \begin{matrix} \Delta \phi = 0, \;\;\; -h \lt z \leq 0, \end{matrix} }[/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] [math]\displaystyle{ \beta \partial_x^4\partial_z \phi - \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\; z = 0, \;\;\; -L \leq x \leq L, }[/math]

where [math]\displaystyle{ \alpha = \omega^2 }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] are the stiffness and mass constant for the plate respectively. The free edge conditions at the edge of the plate imply

[math]\displaystyle{ \partial_x^3 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math] [math]\displaystyle{ \partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math]