Periodic Permanent Roll Waves

Two theories for periodic permanent roll waves are presented which are based on the shallow-water wave equations for the gradually varied portion of the wave profile, and the shock conditions for the rapidly varied portion. In the first theory the channel slope, S₀, is assumed to be sufficiently small that the weight of the shock can be neglected in the shock condition. Comparison of results from this theory for small S₀ with experimental results reveals good agreement for S₀ = 0.019, but rather large discrepancies for S₀ = 0.050, 0.084, and 0.12. In the second theory the shock weight is included in the shock condition. Using the measured shock profiles to evaluate the weight of the shock yields theoretical predictions which are in substantial agreement with the experimental results.