= Solution
Write $e=E'(1)>0$, $q_1=q'(1)>0$ 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> $\epsilon=0$ for the fast <water> and suspension response, retaining the small <friction> parameter $\delta$. For $k\ne0$, the linearized equations are
$$
\begin{aligned}
&H+U=0,\\
&[ik(F^2-1)+3\delta]U=-ikS,\\
&(\gamma+ik)C=\gamma eU,\\
&\lambda S+ikq_1U=\beta(C-eU).
\end{aligned}
$$
Here surface height is $H+S$, and <linearization> of $1-u^2/h$ gives $H-2U=-3U$. Eliminating $H,C,U$ gives the <dispersion relation>
$$
\boxed{\lambda(k)=-\frac{k^2\left(q_1+\dfrac{\beta e}{\gamma+ik}\right)}{3\delta+ik(F^2-1)}.}
$$
In particular, putting $B=F^2-1\ne0$ and expanding at large $|k|$,
$$
\boxed{\lambda(k)=i\frac{q_1}{B}k+\frac{\beta e}{B}-\frac{3\delta q_1}{B^2}+O(k^{-1}).}
$$
Thus the temporal growth rate tends to a finite constant, while the <phase velocity> is
$$
\boxed{v_{\mathrm{ph}}=-\frac{\operatorname{Im}\lambda}{k}\longrightarrow-\frac{q_1}{F^2-1}.}
$$
For <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 $H+S\simeq-F^2U$ is in phase with the bed perturbation in this regime.
For clarity, neglecting $\delta$ completely gives the exact leading-order expressions
$$
\operatorname{Re}\lambda=\frac{\beta e}{F^2-1}\frac{k^2}{\gamma^2+k^2},\qquad v_{\mathrm{ph}}=-\frac1{F^2-1}\left(q_1+\frac{\beta e\gamma}{\gamma^2+k^2}\right).
$$
These also identify the missing qualification in the source: \b[the unstable upstream branch requires $F>1$.] For $F<1$ this leading model gives decay and downstream propagation. The near-critical case $F^2-1\simeq0$ needs a different balance; it is not covered by the expansion. Small nonzero $\epsilon$ 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.
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