Dispersion Relation for a Floating Elastic Plate
Separation of Variables
The dispersion equation arises when separating variables subject to the boundary conditions for a Floating Elastic Plate for a Single Frequency of infinite extent. The equations are described in detail in the Floating Elastic Plate page and we begin with the equations The equations of motion for the Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] is
[math]\displaystyle{ D\frac{\partial^4 \eta}{\partial x^4} - \omega^2 \rho_i h \eta = \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, z=0 }[/math]
plus the equations within the fluid
[math]\displaystyle{ \nabla^2\phi =0 }[/math]
[math]\displaystyle{ g }[/math] is the acceleration due to gravity, [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] the thickness and flexural rigidity of the plate.
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-h }[/math] 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.
