Difference between revisions of "Dispersion Relation for a Floating Elastic Plate"

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= Separation of Variables =
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The dispersion equation for a [[Floating Elastic Plate]] arises when separating
 
The dispersion equation for a [[Floating Elastic Plate]] arises when separating
 
variables subject to the boundary conditions for a [[Floating Elastic Plate]]
 
variables subject to the boundary conditions for a [[Floating Elastic Plate]]
 
of infinite extent.  
 
of infinite extent.  
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The equations of motion for the
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[[Frequency Domain Problem]] with frequency <math>\omega</math> is
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<math>D\frac{\partial^4 \eta}{\partial x^4} - \omega^2 \rho_i h \eta =
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\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in P</math>
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<math>0=
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\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>
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plus the equations within  the fluid
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<math>\nabla^2\phi =0 </math>
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<math> g </math> is the acceleration due to gravity,  <math> \rho_i </math> and <math> \rho </math>
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are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
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the thickness and flexural rigidity of the plate.
  
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<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
 
The (nondimensional) dispersion relation for a [[Floating Elastic Plate]] can be written
 
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  
 
in a number of forms. One form, which has certain theoretical and practical advantages is  

Revision as of 09:29, 12 May 2006

Separation of Variables

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 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, \, x\in P }[/math]

[math]\displaystyle{ 0= \rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F }[/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.