Kinematic Wave Approach to Hydraulic Jumps with Waves

Rayleigh's classical approach to hydraulic jumps is discussed and extended to allow for the added presence of finite amplitude disturbance waves. In this paper the decrease of mean flow energy through the jump discontinuity is accounted for by the sudden appearance of enhanced downstream radiating waves. Without appealing to turbulent dissipation, the proposed model, which applies only to weak inviscid bores, relates the mean height, the mean speed, the wave energy density, and the wavenumber fore and aft of the jump through the conservation laws for total mass, total momentum, total energy, and wave crest number. The model, in shallow water, completely describes the undulatory character of the wavy downstream flow, yielding results in agreement with observed features of bores in nature. "Deep water jumps," although somewhat speculative, are also briefly described, with the aim of stimulating further discussion. The relationships between the present hydraulic model and some work of Benjamin and Lighthill and others are also cited for completeness. The basic notions are then extended to "wave discontinuities" in arbitrary continuous media using ideas derived from kinematic wave theory, and a further stability analysis points to the conditions under which the postulated jumps exist.