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

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The dispersion equation arises when separating
 
The dispersion equation 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]]
for a [[Single Frequency]]
+
of infinite extent.  
of infinite extent. The equations are described in detail in the [[Floating Elastic Plate]]
+
The same equation arises when separating variables in two or three dimensions
 +
and we present here the two-dimensional version.
 +
The equations are described in detail in the [[Floating Elastic Plate]]
 
page and we begin with the equations  
 
page and we begin with the equations  
 
The equations of motion for the
 
The equations of motion for the
[[Frequency Domain Problem]] with frequency <math>\omega</math> is  
+
[[Frequency Domain Problem]] with frequency <math>\omega</math> in terms of
 +
the potential alone is  
  
<math>D\frac{\partial^4 \eta}{\partial x^4} - \omega^2 \rho_i h \eta =  
+
<math>D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z}
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, z=0</math>
+
- \omega^2 \rho_i h \frac{\partial \phi}{\partial z} =  
 +
i\omega \rho g \frac{\partial \phi}{\partial z} - \omega^2 \rho \phi, \, z=0</math>
  
 
plus the equations within  the fluid  
 
plus the equations within  the fluid  

Revision as of 09:45, 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 same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. 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] in terms of the potential alone is

[math]\displaystyle{ D\frac{\partial^4}{\partial x^4} \frac{\partial \phi}{\partial z} - \omega^2 \rho_i h \frac{\partial \phi}{\partial z} = i\omega \rho g \frac{\partial \phi}{\partial z} - \omega^2 \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.