Multiple scales analysis of the undular hydraulic jump over horizontal surfaces

The basic equations for near‐critical stationary turbulent open‐channel flow over horizontal surfaces are solved by means of a multiple scales analysis. The multiple scales solution describing the first‐order perturbation of the free‐surface elevation is compared with the numerical solution of an extended steady‐state version of the Korteweg–de Vries (KdV) equation, confirming the uniform validity of this non‐linear third‐order ODE. In comparison with experimental data the numerical solution of the extended KdV equation shows strong dependence on the initial curvature in terms of wavelength and amplitude.