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A bedform is organized relief on a mobile sediment bed, such as a dune or an antidune. Its shape changes the overlying flow, which changes the shear stress and the sediment transport; erosion and deposition then feed back on the shape. The pattern is therefore a coupled fluid-and-grain response rather than simply a frozen pile of material.

For a small sinusoidal bed disturbance, an upstream lead of the forcing relative to bed height can favor growth, while sediment transport saturation introduces a downstream lag. The Exner equation converts the resulting flux divergence into bed-height change. Their combination determines both growth and migration; the direction of migration cannot be inferred from a bed's appearance alone. Recording a bed profile at successive times separates changing amplitude from movement of a crest, while flow measurements test the assumed bed shear response.

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.

Eliminate the transport amplitudes in the linear stability analysis. For a general bed shear response $\mathcal{T}(k)$, the dispersion relation isThis is the most specific result available without adding the missing hydrodynamic closure. With $\mathcal{T}(k)={\tau }_{0}k(A+iB)$, setMultiplication by $1-ix$ gives the growth rate and migration velocityThe packed-bed volume fraction cancels. For a real physical closure the negative wavenumber response is the complex conjugate; the displayed $k>0$ formulas cover the independent normal modes.

If $A>0$, $B>0$ are treated as constants, the upstream shear stress phase lead promotes growth, while the inclined-bed sediment threshold reduces it and the finite saturation length delays sediment transport. There is an unstable band precisely when $b>0$: $0<k<b/(a{L}_{\mathrm{sat}})$. Within this band $c>0$, so the growing bedforms migrate downstream. The fastest normal mode satisfiesThis last maximization assumes constant $A,B$; if the hydrodynamic coefficients vary with wavenumber, their derivatives also enter the selection condition. When $b\le 0$ and $a>0$, all nonzero normal modes decay.

Just above the sediment entrainment threshold,Thus the common regime $B<1/{\mu }_s$ is stable near onset: the slope penalty overwhelms the hydrodynamic phase lead. This conclusion is conditional on that inequality, not universal. For constant $B>0$, instability begins only when ${\tau }_0>{\tau }_{\mathrm{th},0}/({\mu }_{s}B)$ as well as ${\tau }_0>{\tau }_{\mathrm{th},0}$. The transport sensitivity $C=\chi \gamma {\Delta }^{\gamma -1}$ vanishes at onset for $\gamma >1$, is finite for $\gamma =1$, and is singular for $0<\gamma <1$. Exactly at threshold, the positive-part transport law needs separate treatment; the above linearization with perturbations small compared with $\Delta$ is unavailable.

Far above the sediment entrainment threshold, $b\simeq {\tau }_{0}B$. If $A,B>0$, the unstable cutoff tends to $B/(A{L}_{\mathrm{sat}})$, and the dominant wavelength is set by the saturation length. Both growth rate and migration velocity scale with $C{\tau }_0\sim \chi \gamma {\tau }_0^{\gamma }$, apart from their length factors. Their numerical values and any detailed dependence on wavenumber still require a flow closure.

For example, prescribing uniform shear stress independently of the bed gives $\mathcal{T}=0$ andThis equally admissible closure illustrates why instability cannot be asserted from the printed sediment equations alone.

The plots use constant $a>0$ and three illustrative ratios $b/a$. They show how the sign of the shear phase lead minus the slope correction determines stability; they are not numerical predictions for unspecified flow conditions.

Sediment transport saturation is relaxation of a transported grain population toward the equilibrium sediment flux supported by local flow conditions. A change in bed shear stress does not immediately give a fully adjusted grain speed and population. Close to equilibrium, a steady model isFor constant $q_{\mathrm{sat}}$ and saturation length $L_{\mathrm{sat}}>0$, this gives $q=q_{\mathrm{sat}}+[q(0)-q_{\mathrm{sat}}]e^{-x/L_{\mathrm{sat}}}$. The spatial lag matters for bedform growth.^[ref]

A published wind tunnel test used a $1\mathrm{m}$ wide, $0.5\mathrm{m}$ high and $4.5\mathrm{m}$ long tunnel. Repeated measurements of bed elevation after stopping the flow supplied an erosion rate; integrating that rate inferred downstream flux by mass conservation. The fitted near-equilibrium relaxation length was distinguished from the earlier grain-ejection stage, so the total development distance was not treated as the same parameter.^[ref]