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

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 Relation for a Floating Elastic Plate.