The hydraulic jump calculator above handles the abrupt transition where fast shallow water becomes slow deep water. It is not a gradual change: the surface rises over a distance of a few depths, the flow rolls back on itself, and a great deal of energy is destroyed in the process. Below a spillway, downstream of a sluice gate, at the base of a chute, this is the mechanism that stops the discharge from tearing the river bed apart.
Arb Digital publishes free physics calculators that each own one job, and four of them share the word "hydraulic" while meaning four different things. This page owns the jump itself, an event rather than a property: a supercritical-to-subcritical transition with a specific downstream depth and a specific energy loss. The hydraulic radius calculator owns A divided by P, a geometric property of a channel cross-section. The hydraulic conductivity calculator owns K, how readily water moves through soil. The hydraulic gradient calculator owns i, the dimensionless slope that drives groundwater flow. And the live hydraulic cylinder force calculator shares the word and none of the physics: it is fluid power, computing actuator force from bore and pressure.
What This Hydraulic Jump Calculator Does
You give it the supercritical depth arriving at the jump and one measure of how fast that water is moving — velocity, discharge per unit width or Froude number, whichever you have. It returns the sequent depth on the far side, the Froude numbers either side, the energy head destroyed, the downstream velocity, an estimate of the jump length and the power the jump is dissipating.
It also classifies the jump. The behaviour of a jump changes character with the upstream Froude number, and the standard bands — undular, weak, oscillating, steady and strong — describe genuinely different phenomena, not just different magnitudes of the same one. An oscillating jump throws a wave train downstream that can erode banks a long way from the structure; a steady jump does not.
The nearby open channel flow calculator solves the Manning discharge problem for a channel and reports the Froude number as one of its outputs, which is how you find out that a jump is going to happen. This page is what you use once you know it will.
How to Use It
- Enter the upstream depth. It must be the shallow supercritical depth before the jump, not the deep water after it. Entering the downstream depth gives a Froude number below one and no jump.
- Give one measure of the flow. Velocity if you have measured it, discharge per unit width if you know the flow, or the Froude number directly if a previous calculation produced it.
- Add the channel width if you want power. The jump relations are per unit width, but total discharge and dissipated power need the width.
- Read the classification, not just the numbers. The Froude band tells you what kind of jump it is, and that determines whether a plain apron will do or a baffled basin is needed.
- Treat the length as an estimate. Jump length is empirical, roughly six times the sequent depth, and real values scatter around that.
The Formula: How A Hydraulic Jump Is Calculated
The upstream Froude number is Fr1 = v1 ÷ √(gy1). Applying the momentum equation across the jump in a horizontal rectangular channel gives the sequent depth ratio, sometimes called the Bélanger equation:
y2 ÷ y1 = ½ × [√(1 + 8Fr1²) − 1]
Continuity then gives v2 = v1y1 ÷ y2, and the energy head lost across the jump has the compact closed form
ΔE = (y2 − y1)³ ÷ (4y1y2)
which can be checked against the difference of the two specific energies, E = y + v²/2g. The Bureau of Reclamation's Water Measurement Manual is the standard open reference for open-channel measurement and structures of this kind, and MIT OpenCourseWare's Transport Processes in the Environment, from Civil and Environmental Engineering, covers the momentum and energy framework the derivation sits in.
Work the defaults by hand. With y1 = 0.2 m and v1 = 6 m/s, Fr1 = 6 ÷ √(9.81 × 0.2) = 6 ÷ 1.4007 = 4.284. Then 8Fr1² = 146.79, so √147.79 = 12.157, minus one is 11.157, halved is 5.578, and y2 = 0.2 × 5.578 = 1.1157 m. The downstream velocity is 6 × 0.2 ÷ 1.1157 = 1.0756 m/s. The energy loss is (0.9157)³ ÷ (4 × 0.2 × 1.1157) = 0.7678 ÷ 0.8925 = 0.860 m. Cross-checking, E1 = 0.2 + 36/19.62 = 2.0349 m and E2 = 1.1157 + 1.1568/19.62 = 1.1746 m, and the difference is 0.860 m — the same figure. That is 42.3 per cent of the incoming energy destroyed in a few metres of channel.
Momentum, Not Energy: Why The Bernoulli Equation Fails Here
Almost every open-channel transition is solved with energy. A hydraulic jump cannot be, and understanding why is the single most useful thing on this page.
Inside a jump the flow is violently turbulent, with a roller of recirculating water riding on the surface and large amounts of air entrained. Energy is being converted to heat at a substantial rate, and there is no way to write that loss down in advance. So the energy equation has an unknown loss term and cannot be solved.
Momentum can. The forces acting on the control volume containing the jump are the hydrostatic pressure forces on the two faces and the bed friction, and over the short length of a jump the friction is negligible. So momentum is conserved even though energy is not, and the momentum equation closes with no unknown term. That is where the sequent depth relation comes from, and it is why the answer is a momentum result. Having solved for the depths, the energy loss then falls out as a leftover. The Bernoulli equation calculator handles the smooth transitions where energy conservation does apply, and this page handles the one place it does not.
The Five Jump Types Behave Differently
The Bureau of Reclamation's classification by upstream Froude number is not a matter of degree; the bands describe distinct behaviour.
Between 1 and 1.7 the jump is undular: the surface ripples rather than breaking, and very little energy is lost. Between 1.7 and 2.5 it is a weak jump, with a series of small rollers and a smooth downstream surface. Between 2.5 and 4.5 comes the oscillating jump, which is the awkward one: the entering jet oscillates between the bed and the surface and throws an irregular wave train a long way downstream, and those waves erode banks well beyond the structure. Between 4.5 and 9.0 is the steady jump, well balanced and insensitive to tailwater, which is what designers aim for. Above 9.0 the jump is strong and rough, dissipating up to eighty-five per cent of the incoming energy but with a violent, spray-throwing surface.
The design implication is direct. A structure that puts a jump in the 2.5 to 4.5 band is asking for downstream erosion, and the usual response is to change the geometry so the jump falls in the steady band instead, or to add baffle blocks that force the issue. The Froude number calculator handles the dimensionless group on its own, across free-surface problems generally.
Tailwater Decides Where The Jump Sits
The calculation above gives the depth the jump must produce. Whether the jump actually forms in the place you want depends on whether the channel downstream can supply that depth, and that is a separate question the sequent-depth relation does not answer.
If the natural downstream depth equals the sequent depth, the jump sits stationary right where the supercritical flow arrives. If the downstream depth is greater, the jump is pushed upstream and may drown the outlet, submerging it and cutting the energy dissipation. If the downstream depth is less, the jump is swept downstream, and the fast supercritical flow runs on over unprotected bed until it finds enough depth — which is exactly the scour failure that stilling basins exist to prevent.
This is why a stilling basin is often built with its floor below the downstream bed level. Lowering the floor raises the depth available at the jump relative to the incoming flow, keeping the jump inside the paved basin across the whole range of discharges the structure has to pass. Since tailwater depth varies with discharge and the required sequent depth varies differently, the two curves have to be compared across the full operating range rather than at one design flow. The flow rate calculator and the open channel flow calculator cover the discharge side of that comparison.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering the downstream depth as y₁ — the upstream depth is the shallow supercritical one. Putting the deep side in gives a Froude number below one and no jump at all.
- Using the energy equation across the jump — energy is not conserved and the loss is not known in advance. Only the momentum equation closes, which is why the sequent depth relation looks the way it does.
- Calling the sequent depth a conjugate of specific energy — the two depths of a jump share momentum, not specific energy. The alternate depth at the same energy is a different quantity entirely.
- Ignoring tailwater — the calculation gives the depth the jump requires, not whether the channel can supply it. Too little and the jump sweeps out, too much and it drowns.
- Applying these relations to a sloping or non-rectangular channel — the Bélanger equation assumes a horizontal rectangular bed. Sloping aprons and trapezoidal sections need their own momentum balance.
Related Free Tools From Arb Digital
To find out whether a jump will occur at all, use the open channel flow calculator, which returns the Froude number along with the discharge, or the Froude number calculator for the dimensionless group on its own. The hydraulic radius calculator gives the channel geometry those calculations need. The Bernoulli equation calculator covers the smooth transitions where energy is conserved, and the flow rate calculator the volumetric arithmetic. For groundwater rather than surface water, see the hydraulic conductivity calculator and the hydraulic gradient calculator, and for closed conduits the pipe flow calculator and the Reynolds number calculator. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It is an abrupt transition in an open channel where fast, shallow supercritical flow becomes slow, deep subcritical flow over a very short distance. The surface rises sharply, a turbulent roller forms, and a large fraction of the flow's energy is converted to heat and turbulence.
It is the depth on the downstream side of a hydraulic jump that has the same momentum flux as the supercritical depth upstream. It is given by the Bélanger equation, which is half of the square root of one plus eight times the upstream Froude number squared, minus one, all multiplied by the upstream depth.
Because the energy loss inside the jump is not known in advance, so the energy equation cannot be solved. Momentum is conserved because the only significant forces are the hydrostatic pressures on the two faces, and bed friction is negligible over the short length of a jump.
It depends strongly on the upstream Froude number. A weak jump at Froude 2 loses only a few per cent, a jump at Froude 4.3 loses about forty per cent, and a strong jump above Froude 9 can destroy up to eighty-five per cent of the incoming energy.
Only when the arriving flow is supercritical, meaning its Froude number exceeds one, and the downstream conditions require subcritical flow. If the flow is already subcritical there is nothing to jump from, and the transition simply does not happen.
Roughly six times the sequent depth for the common jump types, though the figure is empirical and scatters. It is used to size stilling basins, and the basin must be long enough to contain the whole jump or the turbulence continues over unprotected bed.
It is the jump type that occurs between Froude numbers of about 2.5 and 4.5, where the entering jet swings between the bed and the surface and sends an irregular wave train downstream. Those waves erode banks well beyond the structure, so designers avoid this band where they can.
Yes, it decides where the jump sits. Too little downstream depth sweeps the jump out of the basin and lets supercritical flow run over unprotected bed; too much drowns the jump and reduces its dissipation. The sequent depth calculation says what is required, not what is available.
This tool is provided for educational and study use. It implements the classical Bélanger momentum relation for a horizontal rectangular channel and does not account for sloping beds, non-rectangular sections, air entrainment, submerged jumps, baffle blocks or scour, so treat its output as a physics result rather than a design-grade figure for any hydraulic structure.
