= Solution
Write $u=1+\widetilde u$, $c=1+\widetilde c$ and let $a_E=E^{*\prime}(1)>0$. To first order $\widetilde s=(1-F^2)\widetilde u$, and the other balances give $\widetilde c_x=a_E\widetilde u-\widetilde c=-\widetilde s_t$. Seek modes $e^{\sigma t+ikx}$. Then
$$
\widetilde c=\frac{a_E}{1+ik}\widetilde u,\qquad
\boxed{\sigma=\frac{a_E}{F^2-1}\frac{ik}{1+ik}=\frac{a_E}{F^2-1}\frac{k^2+ik}{1+k^2}.}
$$
Thus every nonzero <wavenumber> grows when $F>1$, while it decays when $F<1$. The zero mode is neutral in this reduced model. The exceptional $F=1$ limit makes the algebraic depth-bed relation degenerate and requires neglected effects. If the <entrainment> <derivative> vanishes at the reference state, this first-order instability calculation is itself degenerate.
For $F>1$, the <growth rate> tends to the finite positive limit $a_E/(F^2-1)$ as $|k|\to\infty$. It does not diverge with <wavenumber>, although this reduced theory selects no finite fastest scale and should not be extrapolated beyond its long-wave regime. The <phase speed> is
$$
\boxed{c_{\rm phase}=-\operatorname{Im}\sigma/k=-\frac{a_E}{(F^2-1)(1+k^2)}<0.}
$$
The unstable bedforms migrate upstream against the water flow and represent <antidunes> arising from coupled erosion, deposition and supercritical free-surface flow. This is the <relaxation-controlled antidune instability>.
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