Difference between revisions of "Free-Surface Green Function for a Floating Elastic Plate"
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This is a special version of the free-surface Green function which applied when the [[Floating Elastic Plate]] | This is a special version of the free-surface Green function which applied when the [[Floating Elastic Plate]] | ||
− | boundary condition applies at the free-surface. | + | boundary condition applies at the free-surface |
+ | |||
+ | = Two Dimensions = | ||
+ | |||
+ | They first define a function <math>\chi(x,z)</math> representing | ||
+ | outgoing waves as <math>|x|\rightarrow \infty</math> which satisfies | ||
+ | |||
+ | <math> | ||
+ | (\nabla^2 - k_y^2)\chi = 0, -h<z<0, -\infty<x<\infty,{eq:chi1} | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | {\frac{\partial\chi}{\partial z} } =0, z=-h, -\infty<x<\infty, | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | {\left( \beta \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 - | ||
+ | \gamma\alpha + 1\right)\frac{\partial \chi}{\partial z} - \alpha\chi } | ||
+ | = \delta(x), z=0,-\infty<x<\infty,{L} | ||
+ | </math> | ||
+ | |||
+ | This problem can be solved to give | ||
+ | |||
+ | <math> | ||
+ | \chi(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k(n)h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-\kappa(n)|x|}, | ||
+ | </math> | ||
+ | |||
+ | where | ||
+ | |||
+ | <math> | ||
+ | C_n=\frac{1}{2}\left(h + \frac{(5\beta k(n)^4 + 1 - \alpha\gamma)\sin^2{(k(n)h)}}{\alpha}\right), | ||
+ | </math> | ||
+ | |||
+ | and <math>k(n)</math> are the solutions of the [[Dispersion Equation for a Floating Elastic Plate]]. |
Revision as of 10:11, 30 May 2006
This is a special version of the free-surface Green function which applied when the Floating Elastic Plate boundary condition applies at the free-surface
Two Dimensions
They first define a function [math]\displaystyle{ \chi(x,z) }[/math] representing outgoing waves as [math]\displaystyle{ |x|\rightarrow \infty }[/math] which satisfies
[math]\displaystyle{ (\nabla^2 - k_y^2)\chi = 0, -h\lt z\lt 0, -\infty\lt x\lt \infty,{eq:chi1} }[/math]
[math]\displaystyle{ {\frac{\partial\chi}{\partial z} } =0, z=-h, -\infty\lt x\lt \infty, }[/math]
[math]\displaystyle{ {\left( \beta \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 - \gamma\alpha + 1\right)\frac{\partial \chi}{\partial z} - \alpha\chi } = \delta(x), z=0,-\infty\lt x\lt \infty,{L} }[/math]
This problem can be solved to give
[math]\displaystyle{ \chi(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k(n)h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-\kappa(n)|x|}, }[/math]
where
[math]\displaystyle{ C_n=\frac{1}{2}\left(h + \frac{(5\beta k(n)^4 + 1 - \alpha\gamma)\sin^2{(k(n)h)}}{\alpha}\right), }[/math]
and [math]\displaystyle{ k(n) }[/math] are the solutions of the Dispersion Equation for a Floating Elastic Plate.