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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.

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.

Sediment transport is movement of grains by a flowing fluid, through motion close to a bed or within a particle suspension. A flow's shear stress and fluctuating forces compete with grain weight, contacts and friction. The Shields parameter compares a bed-stress scale with the submerged grain-weight scale and helps organize the threshold for movement.

Bedload transport carries grains along or near the bed, while suspended sediment balances settling against upward mixing and entrainment. The transport budget changes the bed itself: mass conservation in the Exner equation relates downstream flux divergence to erosion or deposition. A flow entering an erodible reach can take a finite distance to establish its transported grain population, described by sediment transport saturation. These couplings make a bed an evolving part of the fluid problem rather than a fixed rough wall. Measuring both transported material and bed-height change tests whether the sediment budget closes.

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]