Relaxation-controlled antidune instability ($\sigma =\frac{E^{\prime }(1)}{F^2-1}\frac{ik}{1+ik}$)

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 $\tilde{s}=(1-F^2)\tilde{u}$ and $(1+ik)\tilde{c}=E^{\prime }(1)\tilde{u}$. Elimination proves the displayed dispersion relation.

For $E^{\prime }(1)>0$, the growth rate is $E^{\prime }(1)k^2/[(F^2-1)(1+k^2)]$, positive exactly on the supercritical side $F>1$ for nonzero $k$. The phase speed is $-E^{\prime }(1)/[(F^2-1)(1+k^2)]$, 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.