= Solution
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> as
$$
D=\frac{Eh^3}{12(1-\nu^2)},
$$
where $E$ 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 $\phi=Ae^{kz}e^{i(kx-\omega t)}$ below the mean surface $z=0$, with $k>0$. This solves the <Laplace equation> and decays at depth. For $\eta=\eta_0e^{i(kx-\omega t)}$, the <kinematic boundary condition> $\eta_t=\phi_z$ gives $A=-i\omega\eta_0/k$. The linear <Bernoulli equation> gives
$$
p=-\rho_w(\phi_t+g\eta)=\rho_w\left(\frac{\omega^2}{k}-g\right)\eta.
$$
Eliminating $p$ proves the <dispersion relation of a floating elastic plate>:
$$
\boxed{\omega^2=\frac{\rho_w gk+Dk^5}{\rho_w+\rho_i hk}.}
$$
Thus the desired <wavenumber> is the positive root of
$$
\boxed{\frac{Eh^3}{12(1-\nu^2)}k^5+(\rho_wg-\rho_i h\omega^2)k-\rho_w\omega^2=0.}
$$
For 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 $\omega^2=(gk+Dk^5/\rho_w)\tanh(kd)/[1+(\rho_i h/\rho_w)k\tanh(kd)]$.
Put $\beta=D/\rho_w$ and $\alpha=\rho_i h/\rho_w$. Differentiation gives the <group velocity>
$$
c_g=\frac{d\omega}{dk}=\frac{g+5\beta k^4+4\alpha\beta k^5}{2\omega(1+\alpha k)^2},\qquad T=\frac{2\pi}{\omega}.
$$
For 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 $c_g\sim2\sqrt{D/m}\,k$. 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-77-flexural-speeds.png]
{title=Group and phase speed for an illustrative elastic ice sheet}
The plot is an original model illustration with $h=1\,\mathrm m$, $E=5\,\mathrm{GPa}$, $\nu=0.3$, $\rho_i=917\,\mathrm{kg\,m^{-3}}$, $\rho_w=1025\,\mathrm{kg\,m^{-3}}$, and $g=9.81\,\mathrm{m\,s^{-2}}$. 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. https://agupubs.onlinelibrary.wiley.com/doi/10.1029/97JC02966[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.
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