The analytic description of hydraulic jump in the linear theory of the shear shallow-water flows

This article is devoted to the analytic description of the hydraulic jump in frames of the linear theory of the shear shallow-water flows. Linearization on the piecewise-constant solution is used, and the solution of linear problem is constructed using the Fourier transform, such as in the classical shock wave theory. It allows the solution to be analyzed analytically, in particular, to predict the absence or presence of the hydraulic jump toe oscillations (in terms of shock wave theory-to separate the cases of stability and neutral stability). The simplified example of flows is analyzed, and the formulas for the mentioned separation are obtained.