 High-wavenumber antidune dispersion

In the fast-water limit of an erodible-bed shallow water equations model, let $e=E^{\prime }(1)>0$, $q_1=q^{\prime }(1)>0$ and $\gamma >0$ be the suspended-load relaxation coefficient. Linearization and elimination give the dispersion relation $\lambda =-k^2[q_1+\beta e/(\gamma +ik)]/[3\delta +ik(F^2-1)]$. Its real part tends to the finite constant in the title, and phase velocity tends to $-q_1/(F^2-1)$. Thus the unstable upstream branch requires supercritical flow and enough sediment feedback to overcome friction; subcritical flow has the opposite propagation sign and decay at leading order. The near-critical limit needs a separate balance.