The principal aim of this chapter is to determine the transition from super-critical to sub-critical channel flow. While this transition could occur smoothly, as depicted in Figure 20.2 (with the flow arrow reversed), typically the transition is in the form of a hydraulic shock wave that is abrupt (that is, occurring over a small downstream interval) and spontaneous (that is, occurring without any prompting). Hydraulic shock waves are distinct from compressive shock waves which rely on the compressibility of water for the restoring force and which travel far faster than hydraulic shocks. Hydraulic shocks are impelled into motion by the greater water depth on the lee side of the shock. Shallow-water hydraulic shocks have been given differing names depending on the circumstance of their occurrence. Typically the name hydraulic jump is reserved for a shock that is stationary with respect to an observer standing on the ground. Hydraulic jumps commonly occur below a dam when water is released from the base of the reservoir by lifting a sluice gate slightly. Close to the gate the water flows rapidly, but then the depth increases abruptly at a fixed distance downstream from the gate as the flow abruptly transitions from super-critical to sub-critical. A hydraulic shock moving up a river due to the change in tide is called a bore or surge wave. Bores form in river channels during rising tide, most notably in the Amazon and Orinoco in South America that are reported to have bores as high as 4 m. A hydraulic shock having no water in front is called a flash flood. A flash flood wave moves at the mean speed of the water behind the wave.

Hydraulic shocks often are unsteady when viewed by an observer standing beside the channel, so we must now consider unsteady (time dependent) flows. The details of this transition are complicated, involving turbulent flow and degradation of mechanical energy within a localized downstream interval. In this chapter we will gloss over these details regarding the structure of the shock and investigate the kinematic and dynamic constraints on the uniform flow on either side of a shock. The specific volume flow, q, is conserved across the hydraulic shock.