Difference between revisions of "Free-Surface Green Function for a Floating Elastic Plate"
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= Two Dimensions = | = Two Dimensions = | ||
− | + | The Green function <math>G(x,z)</math> representing | |
− | outgoing waves as <math>|x|\rightarrow \infty</math> | + | outgoing waves as <math>|x|\rightarrow \infty</math> satisfies |
<math> | <math> | ||
− | + | \nabla^2 G = 0, -h<z<0, | |
</math> | </math> | ||
<math> | <math> | ||
− | + | \frac{\partial G}{\partial z} =0, z=-h, | |
</math> | </math> | ||
<math> | <math> | ||
− | {\left( \beta | + | {\left( \beta \frac{\partial^4}{\partial x^4} - |
− | \gamma\alpha + 1\right)\frac{\partial \chi}{\partial z} - \alpha | + | \gamma\alpha + 1\right)\frac{\partial \chi}{\partial z} - \alpha G} |
− | = \delta(x), z=0, | + | = \delta(x), z=0, |
</math> | </math> | ||
+ | |||
+ | (note we are only considering singularities at the free surface). | ||
This problem can be solved to give | This problem can be solved to give | ||
<math> | <math> | ||
− | + | G(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k_n h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-k_n|x|}, | |
</math> | </math> | ||
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</math> | </math> | ||
− | and <math> | + | and <math>k_n</math> are the solutions of the [[Dispersion Relation for a Floating Elastic Plate]], |
+ | with <math>n=-1,-2</math> corresponding to the complex solutions with positive real part, | ||
+ | <math>n=0</math> corresponding to the imaginary solution with positive imaginary part and | ||
+ | <math>n>0</math> corresponding to the real solutions with positive real part. |
Revision as of 11:19, 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
The Green function [math]\displaystyle{ G(x,z) }[/math] representing outgoing waves as [math]\displaystyle{ |x|\rightarrow \infty }[/math] satisfies
[math]\displaystyle{ \nabla^2 G = 0, -h\lt z\lt 0, }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z} =0, z=-h, }[/math]
[math]\displaystyle{ {\left( \beta \frac{\partial^4}{\partial x^4} - \gamma\alpha + 1\right)\frac{\partial \chi}{\partial z} - \alpha G} = \delta(x), z=0, }[/math]
(note we are only considering singularities at the free surface).
This problem can be solved to give
[math]\displaystyle{ G(x,z) = -i\sum_{n=-2}^\infty\frac{\sin{(k_n h)}\cos{(k(n)(z-h))}}{2\alpha C_n}e^{-k_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 Relation for a Floating Elastic Plate, with [math]\displaystyle{ n=-1,-2 }[/math] corresponding to the complex solutions with positive real part, [math]\displaystyle{ n=0 }[/math] corresponding to the imaginary solution with positive imaginary part and [math]\displaystyle{ n\gt 0 }[/math] corresponding to the real solutions with positive real part.