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Assume small-amplitude motion, a thin uniform elastic plate, no prestress, and inviscid, irrotational deep water. Write the upward displacement as $\eta (x,t)$, the ice mass per unit area as $m={\rho }_{i}h$, and its flexural rigidity aswhere $E$ is Young's modulus and $\nu$ is Poisson's ratio. The plate equation is $m{\eta }_{tt}+D{\eta }_{xxxx}=p$, where $p$ is the perturbation of upward water pressure.

Let the water velocity potential be $\varphi =A{e}^{kz}{e}^{i(kx-\omega t)}$ below the mean surface $z=0$, with $k>0$. This solves the Laplace equation and decays at depth. For $\eta ={\eta }_0{e}^{i(kx-\omega t)}$, the kinematic boundary condition ${\eta }_t={\varphi }_z$ gives $A=-i\omega {\eta }_0/k$. The linear Bernoulli equation givesEliminating $p$ proves the dispersion relation of a floating elastic plate:Thus the desired wavenumber is the positive root ofFor positive $D$ and $\omega$, this root is unique: the polynomial is negative at zero and positive eventually; if its derivative starts negative, that derivative crosses zero only once, at a negative-valued minimum before the positive root. If a finite water depth $d$ is required, the corresponding relation is ${\omega }^2=(gk+D{k}^5/{\rho }_w)\tanh (kd)/[1+({\rho }_{i}h/{\rho }_w)k\tanh (kd)]$.

Put $\beta =D/{\rho }_w$ and $\alpha ={\rho }_{i}h/{\rho }_w$. Differentiation gives the group velocityFor long periods, the surface gravity wave limit gives $c_g\sim gT/(4\pi )$. At sufficiently short periods, plate inertia and bending dominate, $\omega \sim \sqrt{D/m}{k}^2$ and $c_g\sim 2\sqrt{D/m}k$. Consequently the curve has a group-velocity minimum of a flexural-gravity wave between branches that rise toward both long and short periods, within the formal deep-water plate model. Very short waves eventually fall outside the thin-plate approximation.

The plot is an original model illustration with $h=1\mathrm{m}$, $E=5\mathrm{GPa}$, $\nu =0.3$, ${\rho }_i=917\mathrm{kg}{\mathrm{m}}^{-3}$, ${\rho }_w=1025\mathrm{kg}{\mathrm{m}}^{-3}$, and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$. Its two minima occur at different periods.

A wave packet near the group-speed minimum carries energy away relatively slowly, so sustained forcing can concentrate energy in that period band. However, wind generation cannot be decided from $c_g$ alone. A steadily translating pressure pattern resonates through its phase velocity $c_p=\omega /k$; it must reach the minimum phase speed of a flexural-gravity wave. Actual wind-wave growth also requires suitable pressure coupling and enough energy input to overcome damping. A coherent cover suppresses ordinary free-surface wave generation, while incoming ocean swell can still excite bending and break the cover.

For disconnected pancake ice or frazil ice, remove the assumption of coherent elastic bending and retain the effective mass loading as a first conservative approximation. Setting $D=0$ gives ${\omega }^2=gk/[1+({\rho }_{i}h/{\rho }_w)k]$. To describe damping as well, replace the elastic layer by a viscous suspension and match velocity and stress to the water beneath: the resulting viscous layer model of frazil-pancake ice has complex $k$, whose real part gives dispersion and imaginary part gives attenuation. Keller's 1998 two-layer model implements this replacement. Effective viscosity and connectedness must reflect the particular cover; zero rigidity alone does not model attenuation.