Difference between revisions of "Dispersion Relation for a Floating Elastic Plate"
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| − | The (nondimensional) dispersion relation for a | + | 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>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. | + | <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.
