= Dispersion relation of a floating elastic plate
{title2=$\omega^2=\frac{\rho_wgk+Dk^5}{\rho_w+mk}$}
For <bending stiffness> $D$ and mass per area $m$, a <plane wave> of surface displacement $\eta$ satisfies $m\eta_{tt}+D\nabla_H^4\eta=p$. An inviscid deep-water <velocity potential> has vertical dependence $e^{kz}$; the <kinematic boundary condition> and linear <Bernoulli equation> give $p=(\rho_w\omega^2/k-\rho_wg)\eta$. Eliminating $p$ gives the displayed <dispersion relation>. With water depth $d$, replace it by $\omega^2=(gk+Dk^5/\rho_w)\tanh(kd)/[1+(m/\rho_w)k\tanh(kd)]$.
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