Hydraulic Jump Calculator – Free Calculator

Understanding the hydraulic jump calculator — concept and principle

A hydraulic jump is the sudden, turbulent rise in water surface that happens when fast, shallow (supercritical) flow slows to deep, tranquil (subcritical) flow. You see it at the foot of spillways, below sluice gates and in stilling basins built to destroy energy before water reaches an erodible riverbed. This calculator finds the depth after the jump, the upstream Froude number and the energy lost in the jump for a rectangular channel.

The jump turns speed into depth and burns off energy in a churning roller.

How it is calculated

Upstream Froude number: Fr₁ = v₁/√(g·y₁), with g = 9.81 m/s².
Sequent (downstream) depth: y₂ = (y₁/2)·(√(1 + 8Fr₁²) − 1), the Bélanger equation.
Head loss: ΔE = (y₂ − y₁)³/(4·y₁·y₂), in metres of water.

y₁ is the depth just before the jump (m) and v₁ the mean velocity there (m/s).

The principle

Inside a jump, energy is not conserved because large eddies turn mechanical energy into heat, so energy cannot predict y₂. Momentum is conserved instead: between the two sections, the difference in hydrostatic pressure force equals the change in momentum flux, with wall friction neglected over the short jump length. Writing this balance per unit width with continuity (q = v₁y₁ = v₂y₂) gives a quadratic in y₂ whose positive root is the Bélanger equation. Substituting y₂ back into the specific energy E = y + v²/(2g) on each side gives the compact loss formula.

Worked example

Inputs: y₁ = 0.3 m, v₁ = 6 m/s.
√(9.81 × 0.3) = 1.7155 m/s, so Fr₁ = 6/1.7155 = 3.497.
√(1 + 8 × 3.497²) = 9.9428, so y₂ = 0.15 × (9.9428 − 1) = 1.341 m (about 4.47 times y₁).
ΔE = (1.341 − 0.3)³/(4 × 0.3 × 1.341) = 0.702 m.
Check: E₁ = 0.3 + 6²/19.62 = 2.135 m, v₂ = 1.8/1.341 = 1.342 m/s, E₂ = 1.433 m, and E₁ − E₂ = 0.702 m.

Each point is the calculator re-run with only “Upstream velocity (m/s)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Upstream velocity (m/s)	Downstream depth (m)
1.5	0.2501433229
3	0.6069403646
4.5	0.9729568603
6	1.341420418
7.5	1.710878011
9	2.080835817
12	2.821504577

How to read the result

The Froude number classifies the jump. Following the US Bureau of Reclamation: about 1–1.7 undular, 1.7–2.5 weak, 2.5–4.5 oscillating, 4.5–9 steady and above 9 strong. The example (Fr₁ ≈ 3.5) is oscillating, which sends waves downstream and is harder to contain; steady jumps are preferred in stilling basins. The jump forms where the actual tailwater depth matches y₂. If tailwater is lower, the jump moves downstream and may scour the bed.

Common mistakes

Applying the formula when Fr₁ ≤ 1; a jump only forms from supercritical flow.
Using it for non-rectangular or steep channels without correction.
Expressing energy loss as a percentage without first computing E₁.

Keep in mind: These equations assume a horizontal, rectangular, smooth channel. Stilling basin design needs tailwater analysis, model studies or established design standards and review by a qualified hydraulic engineer.

Frequently asked questions

How long is a hydraulic jump?

For steady jumps the length is roughly six times the downstream depth, which guides basin length.

What is the energy efficiency of a jump?

E₂/E₁; in the example about 67% of the energy remains and 33% is dissipated.

Why is it called the sequent or conjugate depth?

y₁ and y₂ are the two depths with the same momentum function for the same flow rate.

Sources: V. T. Chow, Open-Channel Hydraulics, McGraw-Hill, 1959 · A. J. Peterka, Hydraulic Design of Stilling Basins and Energy Dissipators, US Bureau of Reclamation Engineering Monograph No. 25, 1958

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

Your numbers