# Template:Linear elastic plate on water time domain

We begin with the linear equations for a fluid. The kinematic condition is the same

$\displaystyle{ \frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; }$

but the dynamic condition needs to be modified to include the effect of the the plate

$\displaystyle{ \rho g\zeta + \rho \frac{\partial\Phi}{\partial t} = D \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} , \ z=0; }$

We also have Laplace's equation

$\displaystyle{ \Delta \Phi = 0,\,\,-h\lt z\lt 0 }$

and the usual non-flow condition at the bottom surface

$\displaystyle{ \partial_z \Phi = 0,\,\,z=-h, }$

where $\displaystyle{ \zeta }$ is the surface displacement, $\displaystyle{ \Phi }$ is the velocity potential, and $\displaystyle{ \rho }$ is the fluid density.