Dispersion Relation for a Floating Elastic Plate

The dispersion equation for a Floating Elastic Plate arises when separating variables subject to the boundary conditions for a Floating Elastic Plate of infinite extent.

The (nondimensional) dispersion relation for a Floating Elastic Plate can be written in a number of forms. One form, which has certain theoretical and practical advantages is the following,

[math]\displaystyle{ f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0, }[/math]

where [math]\displaystyle{ H }[/math] is the nodimensional water depth, and

[math]\displaystyle{ \varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2). }[/math]

[math]\displaystyle{ k }[/math] is the waver number for a wave of radial frequency [math]\displaystyle{ omega }[/math] traveling in open water of infinite depth, [math]\displaystyle{ g }[/math] is the acceleration due to gravity, [math]\displaystyle{ \sigma }[/math] is the amount of the plate that is submerged, [math]\displaystyle{ \rho_i }[/math] and [math]\displaystyle{ \rho }[/math] are the densities of the plate and the water respectively, [math]\displaystyle{ h }[/math] and [math]\displaystyle{ D }[/math] are the thickness and flexural rigidity of the plate, and [math]\displaystyle{ L }[/math] is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber [math]\displaystyle{ gamma/L }[/math] and thus wave speed to the above parameters.