Dispersion Relation for a Floating Elastic Plate
Dispersion Relation for a Floating Elastic Plate
The (nondimensional) dispersion relation for a floating thin elastic plate can be written
[math]\displaystyle{ f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0, }[/math]
where
[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 }[/math] and thus speed ([math]\displaystyle{ =omega/gamma }[/math]) to the above parameters.
