Free-Surface Green Function for a 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

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.