= High-wavenumber antidune dispersion
{title2=$\lambda\sim iq_1k/(F^2-1)+\beta e/(F^2-1)-3\delta q_1/(F^2-1)^2$}
In the fast-<water> limit of an erodible-bed <shallow water equations> model, let $e=E'(1)>0$, $q_1=q'(1)>0$ 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.
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