 Solution

Write $u=1+\tilde{u}$, $c=1+\tilde{c}$ and let $a_E=E^{\ast \prime }(1)>0$. To first order $\tilde{s}=(1-F^2)\tilde{u}$, and the other balances give ${\tilde{c}}_x=a_E\tilde{u}-\tilde{c}=-{\tilde{s}}_t$. Seek modes $e^{\sigma t+ikx}$. ThenThus 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 isThe 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.