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

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The (nondimensional) dispersion relation for a floating thin elastic plate can be written
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The dispersion equation for a [[Floating Elastic Plate]] arises when separating
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variables subject to the boundary conditions for a [[Floating Elastic Plate]]
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of infinite extent.
 +
 
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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>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>
 
<math>f(\gamma)=\cosh(\gamma H)-(\gamma^4+\varpi)\gamma\sinh(\gamma H)=0,</math>
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<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>
 
<math>\varpi=(1-k\sigma)/(kL),\quad k=\omega^2/g,\quad\sigma=\rho_ih/\rho,\quad L^5=D/(\rho\omega^2).</math>
  
<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> sigma </math> is the amount of the plate that is submerged, <math> rho_i </math> and <math> rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.
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<math> k </math> is the waver number for a wave of radial frequency <math> omega </math> traveling in open water of infinite depth, <math> g </math> is the acceleration due to gravity, <math> \sigma </math> is the amount of the plate that is submerged, <math> \rho_i </math> and <math> \rho </math> are the densities of the plate and the water respectively, <math> h </math> and <math> D </math> are the thickness and flexural rigidity of the plate, and <math> L </math> is the natural length that we have scaled length variables by. The dispersion relation relates the wavenumber <math> gamma/L </math> and thus wave speed to the above parameters.

Revision as of 09:25, 12 May 2006

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 (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.