Bedform

Combining the Exner equation, sediment transport saturation, bed shear response, and inclined-bed sediment threshold gives a competition between upstream forcing and downstream relaxation. With $a={\tau }_{0}A$, $b={\tau }_{0}B-{\tau }_{\mathrm{th},0}/{\mu }_s$, $C=\chi \gamma ({\tau }_0-{\tau }_{\mathrm{th},0}{)}^{\gamma -1}$, and positive wavenumber $k$, the growth rate is $\sigma =C{k}^2(b-ak{L}_{\mathrm{sat}})/(1+k^2{L}_{\mathrm{sat}}^2)$. Thus $b>0$ permits long-wave growth when $a>0$, while a finite saturation length stabilizes shorter waves.

For a small sinusoidal bedform, a fluid-dynamical closure can be written $\hat{\tau }={\tau }_{0}k(A+iB)\hat{\eta }$ for $k>0$. The real coefficient $A$ is in phase with the bed height and $B$ describes an upstream phase lead. Neither coefficient follows from the Exner equation or sediment relaxation alone; they require a flow calculation or measurement.