Critical Depth in a Rectangular Channel

Critical depth is the depth at which a given discharge carries the least specific energy it possibly can, and equivalently the depth at which the Froude number is exactly 1. For a rectangular channel it falls out in one line from the unit discharge $q = Q/b$: $y_c = (q^2/g)^{1/3}$. Ten cubic metres a second in a five metre channel gives $q = 2$ m²/s and $y_c = (4/9.807)^{1/3} = 0.742$ m. Notice what is absent. Slope does not appear, roughness does not appear, and critical depth is therefore a property of the channel shape and the flow alone.

That absence is what makes critical depth useful. Any place a channel is forced through critical depth is a control section, a point where depth and discharge are locked to each other, and that is the operating principle behind every flume and broad-crested weir ever built. Measure one depth at a control and you know the flow, with no velocity measurement and no rating curve drift. It is also why the transition from a mild slope to a steep one always passes through critical depth right at the break.

A useful pair of consequences: at critical depth the velocity head is exactly half the depth, so the minimum specific energy of a rectangular channel is $1.5 y_c$, and a bed hump higher than the available energy surplus will choke the flow and force the upstream water level to rise. The mistake to avoid is applying this formula to a trapezoidal or circular section. The general condition is $Q^2 T/(g A^3) = 1$, which for anything but a rectangle has to be solved iteratively, and the rectangular shortcut can be well off on a trapezoid with steep side slopes.