= Relaxation-controlled antidune instability
{title2=$\sigma=\frac{E'(1)}{F^2-1}\frac{ik}{1+ik}$}
In a reduced <sediment>-bed model with normalized discharge one, water momentum gives $s=1-u^{-1}+F^2(1-u^2)/2$, while <concentration> adaptation and bed exchange give $c_x=E(u)-c=-s_t$. Linearizing about $u=c=1,s=0$ yields $\widetilde s=(1-F^2)\widetilde u$ and $(1+ik)\widetilde c=E'(1)\widetilde u$. Elimination proves the displayed dispersion relation.
For $E'(1)>0$, the <growth rate> is $E'(1)k^2/[(F^2-1)(1+k^2)]$, positive exactly on the supercritical side $F>1$ for nonzero $k$. The <phase speed> is $-E'(1)/[(F^2-1)(1+k^2)]$, so unstable bedforms migrate upstream and are <antidunes>. At high <wavenumber> growth approaches a finite plateau rather than becoming unbounded. This closure selects no finite fastest wavelength; short-scale selection needs additional physics. The critical limit $F=1$ and a vanishing <entrainment> <derivative> are degenerate cases outside the displayed conclusion.
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