Difference between revisions of "Dispersion Relation for a Floating Elastic Plate"
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<math>D\frac{\partial^4 \eta}{\partial x^4} - \omega^2 \rho_i h \eta = | <math>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, \, | + | \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, z=0</math> |
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plus the equations within the fluid | plus the equations within the fluid | ||
Revision as of 09:39, 12 May 2006
Separation of Variables
The dispersion equation arises when separating variables subject to the boundary conditions for a Floating Elastic Plate of infinite extent. 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.
