 Solution

Write $e=E^{\prime }(1)>0$, $q_1=q^{\prime }(1)>0$ and introduce perturbations $(H,U,C,S)e^{\lambda t+ikx}$ to water depth, velocity, concentration and bed height. Take the leading quasi-steady approximation $ϵ=0$ for the fast water and suspension response, retaining the small friction parameter $\delta$. For $k\ne 0$, the linearized equations areHere surface height is $H+S$, and linearization of $1-u^2/h$ gives $H-2U=-3U$. Eliminating $H,C,U$ gives the dispersion relationIn particular, putting $B=F^2-1\ne 0$ and expanding at large $|k|$,Thus the temporal growth rate tends to a finite constant, while the phase velocity isFor supercritical flow $F>1$, the waves propagate upstream. They grow if $\beta e(F^2-1)>3\delta q_1$; at leading order in small $\delta$, this holds for fixed positive $\beta ,e$ away from critical flow. They are antidunes, upstream-migrating bedforms coupled to the free surface. Indeed $H=-U$ and $S\simeq -(F^2-1)U$, so surface height $H+S\simeq -F^{2}U$ is in phase with the bed perturbation in this regime.

For clarity, neglecting $\delta$ completely gives the exact leading-order expressionsThese also identify the missing qualification in the source: the unstable upstream branch requires $F>1$. For $F<1$ this leading model gives decay and downstream propagation. The near-critical case $F^2-1\simeq 0$ needs a different balance; it is not covered by the expansion. Small nonzero $ϵ$ perturbs the slow bedform branch and adds fast water modes, so the stated result concerns the leading slow morphodynamic branch. This is the high-wavenumber antidune dispersion calculation.