Open-Channel Specific Energy and Alternate Depth
The specific-energy diagram the Froude tile only names: E = y + q^2/(2 g y^2) is the open-channel flow energy measured from the bed, and for one discharge the same E passes at two alternate depths - a deep tranquil one and a shallow shooting one - that meet at the critical depth where the energy bottoms out at Ec = 1.5 yc. A 10 ft channel at 100 cfs and a 3 ft depth carries 3.17 ft of energy (Fr 0.34, subcritical); that same energy also passes at 0.81 ft (supercritical), and the minimum is 2.19 ft at the 1.46 ft critical depth. The two depths share their energy with no loss, unlike the hydraulic jump's sequent depth, which dissipates it. Rectangular, prismatic channel. A design aid; the engineer of record governs.
Formula and source
q = Q/b; V = q/y; E = y + q^2/(2 g y^2); yc = (q^2/g)^(1/3); Ec = 1.5 yc; Fr = V/sqrt(g y); y_alt = ((E - y) + sqrt((E - y)^2 + 2 q^2/(g y)))/2.
The open-channel specific-energy relation E = y + q^2/(2 g y^2), the rectangular critical depth yc = (q^2/g)^(1/3) with minimum specific energy Ec = 1.5 yc, and the alternate depth sharing the same E, as compiled in Chow's Open-Channel Hydraulics, by name.
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