In this paper the two-dimensional depth-averaged momentum and continuity equations for steady shallow flow and sediment transport are described. This two-dimensional model is used to study of the bed deformation in alluvial channels. The perturbation of mean values in the cross-section of hydraulic flow parameters is determined by the periodic function which describes the non-uniform but steady flow. The function of this type allows the investigation of the flow stability in both: the straight and the curvilinear cannels. For these channels bed topography and horizontal distribution of depth-averaged velocity in longitudinal direction can be computed from exponential-periodic solution of the momentum and continuity equations. Bed topography computed from this solution is compared with bed topography of two bends in the Warta River. It was found that this solution is in good agreement with the measured bed topography in the river. It is also found that the cosine-generated curve is the natural curvature of the river meanders. Two equations for the calculation of the minimum value of meander curvature at the apex and the optimal length of bend are also presented in the paper. ...
