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The movement of bed load along the closed conduit can lead to a loss of stability of the bed surface, when bed waves arise at the bed of the channel. Investigation of the development of bed waves is associated with the possibility of determining of the bed load nature along the bed of the periodic form. Despite the great attention of many researchers to this problem, the question of the development of bed waves remains open at the present time. This is due to the fact that in the analysis of this process many researchers use phenomenological formulas for sediment transport in their work. The results obtained in such models allow only assess qualitatly the development of bed waves. For this reason, it is of interest to carry out an analysis of the development of bed waves using the analytical model for sediment transport.

The paper proposed two-dimensional profile mathematical riverbed model, which allows to investigate the movement of sediment over a periodic bed. A feature of the mathematical model is the possibility of calculating the bed load transport according to an analytical model with the Coulomb–Prandtl rheology, which takes into account the influence of bottom surface slopes, bed normal and tangential stresses on the movement of bed material. It is shown that when the bed material moves along the bed of periodic form, the diffusion and pressure transport of bed load are multidirectional and dominant with respect to the transit flow. Influence of the effects of changes in wave shape on the contribution of transit, diffusion and pressure transport to the total sediment transport has been studied. Comparison of the received results with numerical solutions of the other authors has shown their good qualitative initiation.

In this paper on the basis of the riverbed model proposed earlier the one-dimensional stability problem of closed flow channel with sandy bed is solved. The feature of the investigated problem is used original equation of riverbed deformations, which takes into account the influence of mechanical and granulometric bed material characteristics and the bed slope when riverbed analyzing. Another feature of the discussed problem is the consideration together with shear stress influence normal stress influence when investigating the riverbed instability. The analytical dependence determined the wave length of fast-growing bed perturbations is obtained from the solution of the sandy bed stability problem for closed flow channel. The analysis of the obtained analytical dependence is performed. It is shown that the obtained dependence generalizes the row of well-known empirical formulas: Coleman, Shulyak and Bagnold. The structure of the obtained analytical dependence denotes the existence of two hydrodynamic regimes characterized by the Froude number, at which the bed perturbations growth can strongly or weakly depend on the Froude number. Considering a natural stochasticity of the waves movement process and the presence of a definition domain of the solution with a weak dependence on the Froude numbers it can be concluded that the experimental observation of the of the bed waves movement development should lead to the data acquisition with a significant dispersion and it occurs in reality.

The work is a theoretical study of the development of bottom instability in rivers and canals. Based on an analytical model of the load of sediment, taking into account the influence of slopes of the bottom surface, bottom pressure and shear stress on the movement of the bottom material and an analytical solution that allows to determine bottom tangential and normal stresses over the periodic bottom, the problem of determining the amplitude growth rate for growing bottom waves is formulated and solved . The obtained solution of the problem allows us to determine the characteristic time of the growth of the bottom wave, the growth rate of the bottom wave and its maximum amplitude, depending on the physical and particle size characteristics of the bottom material and the hydraulic parameters of the water flow. On the example of the development of a periodic sinusoidal bottom wave of low steepness, the verification of the solution obtained for the formulated problem is carried out. The obtained analytical solution to the problem allows us to determine the growth rate of the amplitude of the bottom wave from the current value of its amplitude. Comparison of the obtained solution with experimental data showed their good qualitative and quantitative agreement.