= Solution
Write $w(x,t)$ for the vertical displacement of the <sea ice>, and take the undisturbed water surface as $z=0$, with water occupying $z<0$. The <elastic plate> has areal mass $m=\rho_i h$ and <bending stiffness>
$$
D=\frac{Eh^3}{12(1-\eta^2)}.
$$
Here $E$ is <Young's modulus> and $\eta$ is <Poisson's ratio>. We neglect in-plane prestress, viscosity and plate shear deformation, and linearize about hydrostatic equilibrium. These are important assumptions: perfect elasticity alone does not specify every term in a floating-plate model.
For a <plane wave> $w=\widehat w e^{i(kx-\omega t)}$, $k>0$, <potential flow> in deep water has <velocity potential> $\phi=\widehat\phi e^{kz}e^{i(kx-\omega t)}$. This solves <Laplace's equation> and decays downwards. The <kinematic boundary condition> gives $k\widehat\phi=-i\omega\widehat w$. Linearizing the water pressure at the displaced interface gives an upward excess load
$$
\widehat p=-\rho_w(\phi_t+gw)
=\rho_w\left(\frac{\omega^2}{k}-g\right)\widehat w.
$$
The <elastic plate> equation is $m w_{tt}+D w_{xxxx}=p$. Substitution and multiplication by $k$ therefore give the <flexural-gravity wave> <dispersion relation>
$$
\boxed{Dk^5+(\rho_wg-\rho_i h\omega^2)k-\rho_w\omega^2=0.}
$$
Equivalently, with $\alpha=\rho_i h/\rho_w$ and $\beta=D/\rho_w$,
$$
\boxed{\omega^2=\frac{gk+\beta k^5}{1+\alpha k}.}
$$
This determines the positive <wavenumber> implicitly for prescribed positive <angular frequency>. It is unambiguous: the <derivative> of $\omega^2$ is $(g+5\beta k^4+4\alpha\beta k^5)/(1+\alpha k)^2>0$, while $\omega^2$ runs from zero to infinity.
The <phase velocity> and <group velocity> are
$$
c_p=\frac{\omega}{k},\qquad
\boxed{c_g=\frac{g+5\beta k^4+4\alpha\beta k^5}{2\omega(1+\alpha k)^2}.}
$$
For open-water <deep-water gravity waves>, $\omega^2=gk$, and hence $c_p=gT/(2\pi)$ and $c_g=gT/(4\pi)$. At large period the ice-covered curves approach these straight lines. At shorter period, plate bending raises the speeds, so both ice-covered curves turn upward as period decreases. In the bending regime with negligible plate inertia, $\omega\propto k^{5/2}$ and $c_g\simeq(5/2)c_p$; in the formal plate-inertia-dominated limit, $\omega\propto k^2$ and $c_g\simeq2c_p$. The latter extrapolation eventually leaves thin-plate validity and should not be read as a prediction at arbitrarily small <wavelength>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-72-flexural-gravity.png]
{title=Phase and group velocities of flexural-gravity waves compared with open-water waves}
{height=576}
The plotted parameters are illustrative rather than measured at the observation site. They give a <group-velocity minimum of a flexural-gravity wave> near $15.1\,\mathrm{m\,s^{-1}}$ at period $17.6\,\mathrm s$. \b[There is no arbitrarily slow wave-energy branch under a continuous elastic sheet.] Energy put into a localized disturbance travels away at at least this minimum <group velocity>; a slowly moving wind system cannot retain a <wave packet> indefinitely beneath itself. This reduces the opportunity for sustained local growth compared with slow, short open-water waves, and the continuous cover also prevents direct wind forcing of an exposed water surface. Incoming long swell can still propagate.
A <group-velocity minimum of a flexural-gravity wave> is not by itself a universal minimum wind speed for wave generation. A steadily translating forcing pattern requires a <phase velocity> matching its translation speed, so the minimum of $c_p$, not $c_g$, supplies the corresponding resonance threshold. Random wind forcing, dissipation and aerodynamic coupling must be specified before making an absolute generation claim.
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