Energy Loss in a Hydraulic Jump

Subtract the specific energy after a jump from the specific energy before it, substitute the momentum relation between the two depths, and a remarkable amount cancels: the loss is $(y_2-y_1)^3/(4y_1y_2)$. One metre jumping to four metres destroys $27/16 = 1.6875$ m of head. The cube in the numerator is the interesting part, because it means the energy destroyed grows far faster than the depth ratio does. A jump from 0.5 to 2.0 metres, the same fourfold ratio at half the scale, loses only 0.84 m.

That cube is why stilling basins work at all. At $Fr_1 = 2$ a jump kills about 6 percent of the incoming energy, at $Fr_1 = 5$ about half of it, and at $Fr_1 = 10$ around 75 percent. A high-head spillway therefore deliberately generates the most violent jump it can, because every joule dissipated in a concrete basin is a joule that does not scour the river downstream. Making the jump gentler is exactly the wrong instinct.

Two cautions from the field. The energy has to go somewhere, and it goes into turbulence, noise, air entrainment and heat, along with pressure fluctuations that can reach several metres of head at frequencies that have destroyed baffle blocks and lifted floor slabs. Basins are designed with anchorage and drainage for that reason, not just for the geometry. And a jump is a superb aerator, which is why the same phenomenon that wrecks a spillway apron is used deliberately on a cascade to strip carbon dioxide or add oxygen to a stream. On the flip side, the dissolved gas supersaturation downstream of a deep plunge basin can be lethal to fish, which is a documented problem below several Columbia River dams.