Template:Linear elastic plate on water time domain

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

\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

\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

\Delta \Phi = 0,\,\,-h<z<0

and the usual non-flow condition at the bottom surface

\partial_z \Phi = 0,\,\,z=-h,

where ζ is the surface displacement, Φ is the velocity potential, and ρ is the fluid density.

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