The Hydraulic Jump: Where a Sheet of Water Slams to a Halt

The shallow-water sound barrier

The whole phenomenon hangs on one dimensionless number. In a thin layer of water of depth h, small surface disturbances — gravity waves — travel at the shallow-water wave speed c = √(gh), where g = 9.81 m/s². For h = 1 mm that is only c ≈ 0.099 m/s; for h = 3 mm, c ≈ 0.17 m/s. Compare this to the bulk speed U of the water itself through the ratio

• Fr = U / √(gh) — the Froude number (William Froude, 1860s), the shallow-water twin of the Mach number.

When Fr > 1 the flow is supercritical: the water moves faster than its own waves, so no disturbance can propagate upstream. The thin sheet leaving your tap is blind to whatever lies ahead. When Fr < 1 the flow is subcritical and waves can crawl back upstream. A hydraulic jump is the abrupt, dissipative bridge from Fr > 1 to Fr < 1 — the hydraulic analogue of a supersonic shock wave decelerating to subsonic flow. Just as a shock cannot be smooth, neither can the jump: it is a genuine discontinuity a few depths wide.

Why momentum, not energy, sets the height

A beginner's instinct is to apply Bernoulli — but Bernoulli assumes no losses, and a jump is a machine for making losses. The correct conserved quantity across the discontinuity is the flux of momentum plus pressure force. Consider a rectangular channel of unit width, upstream depth h₁ and speed U₁, downstream depth h₂ and speed U₂. Two exact statements hold:

• Continuity (mass): U₁h₁ = U₂h₂ = q, the discharge per unit width (m²/s).
• Momentum: the hydrostatic pressure force plus momentum flux is equal on both sides: ½ρg h₁² + ρ U₁²h₁ = ½ρg h₂² + ρ U₂²h₂.

The ½ρg h² terms are the depth-integrated hydrostatic pressure; the ρU²h terms are the momentum carried by the moving water. Crucially, we do not equate the mechanical energy — that is allowed to drop. This is exactly why a hydraulic jump works as a stilling device: momentum is conserved, energy is spent. The sum M = ½g h² + q²/h is called the specific momentum (or force function), and the two depths that share the same M are the conjugate (or sequent) depths.

The Bélanger equation — solving for the jump

Combine continuity and momentum. Substitute U = q/h and eliminate U₂ using U₂ = U₁h₁/h₂. After dividing through by the common factors, the momentum balance collapses to a quadratic in the depth ratio r = h₂/h₁, whose physical root is the celebrated Bélanger equation (Jean-Baptiste Bélanger, 1828):

• h₂/h₁ = ½ ( √(1 + 8·Fr₁²) − 1 )

Every symbol earns its place: Fr₁ = U₁/√(gh₁) is the upstream Froude number, and the depth ratio depends on nothing else in this idealised balance — not viscosity, not surface tension. A few concrete values:

• Fr₁ = 1: ratio = 1 (no jump — the critical, marginal case).
• Fr₁ = 2: ratio = ½(√33 − 1) ≈ 2.37 — a modest 'undular' or weak jump.
• Fr₁ = 3.5: ratio = ½(√99 − 1) ≈ 4.48 — a strong, fully turbulent jump.
• Fr₁ = 9: ratio ≈ 12.3 — the violent spillway regime.

The same algebra run backwards gives the downstream Froude number Fr₂ = Fr₁ / (½(√(1+8Fr₁²) − 1))^(3/2), which is always < 1 for Fr₁ > 1 — the flow really has crossed from supercritical to subcritical.

Accounting for the missing energy

Because momentum (not energy) is conserved, we can afterwards measure how much energy the jump destroyed. The specific energy — head measured from the channel bed — is E = h + U²/(2g) = h + q²/(2gh²). Subtracting downstream from upstream and simplifying with the Bélanger relation gives a strikingly clean result:

• ΔE = E₁ − E₂ = (h₂ − h₁)³ / (4 h₁ h₂)

Every symbol is a length, so ΔE is a head loss in metres; multiply by ρg to get energy per unit volume (J/m³) or by ρg·q per unit width for power dissipated (W/m). The cubic dependence on (h₂ − h₁) is why engineers love the jump: a strong jump (Fr₁ ≈ 9) dissipates 70–85% of the incoming mechanical energy in a channel only a few metres long. For a modest jump Fr₁ = 2 the loss is only a few percent; the efficiency of destruction rises steeply with Froude number. That head loss does not vanish — it becomes turbulent kinetic energy, then heat, warming the water by a tiny but real amount (a 1 m head loss corresponds to ΔT ≈ g·Δh/c_water ≈ 9.81/4186 ≈ 0.0023 K).

The circular jump in your sink

Your kitchen sink runs the same physics with an added twist: radial geometry and surface tension. A vertical jet of radius a strikes the flat sink and spreads as a thin film. Because discharge Q is fixed, continuity in the film means the radial speed and depth vary with radius r: as the film thins and slows outward, its local Froude number falls. At the radius R where Fr drops through ~1, the film can no longer outrun its own waves and a circular hydraulic jump snaps into being — the bright ring. A useful scaling (Watson, 1964; Bohr, Dimon & Putkaradze, 1993) for the jump radius is

• R ∝ Q^(5/8) ν^(−3/8) g^(−1/8), with Q the volume flow rate and ν the kinematic viscosity (water: ν ≈ 1.0×10⁻⁶ m²/s).

Turn up the flow and the ring grows; the exponent 5/8 means doubling the tap flow pushes the ring out by only ~50%. At these millimetre scales surface tension (γ ≈ 0.072 N/m for water) matters: it adds an extra pressure jump ~γ/R across the curved rim and can select between smooth and polygonal jump shapes (yes, hexagonal and pentagonal jumps are real and photographable). Viscosity, absent from Bélanger's ideal balance, here sets the whole length scale — because the thin film is dominated by the boundary layer growing up from the sink surface.

From spillways to tidal bores

The hydraulic jump is one of civil engineering's workhorses. Below a dam, water accelerates down the spillway to tens of metres per second — supercritical, Fr₁ often 5–12. If that jet reached the river unchecked it would scour the bed to destruction, so engineers build a stilling basin whose baffle blocks force a jump precisely where they want it. The Bureau of Reclamation classifies basins by Fr₁ (Type II, III, IV) using exactly the Bélanger depth ratio to size the tailwater. The jump can turn a 40 m/s torrent into a placid 2–3 m/s river within a basin length of roughly 6h₂, converting gigawatts of would-be erosive power into froth.

• Tidal bores: the wall of water surging up the Severn, Qiantang, or Amazon (the pororoca) at 3–8 m/s is a moving hydraulic jump — an undular or breaking jump propagating upstream against the river.
• Ship wakes and sluice gates: the standing wave downstream of an open gate is a jump; the sharp 'rooster tail' behind fast boats obeys the same Fr criterion.
• Sink and gutter flows: every roadside gutter and dishwasher spray forms them; the atmosphere even has an analogue in the 'Morning Glory' and mountain lee-wave hydraulic jumps of stratified airflow.

Subtleties, classifications, and common myths

Myth 1: 'The jump is where the water speeds up.' Exactly backwards — the water slows and deepens. Kinetic energy converts partly to potential energy (raised surface) and mostly to heat. Myth 2: 'Bernoulli explains it.' Bernoulli fails precisely because a jump is dissipative; only the momentum balance is exact across it. Myth 3: 'A jump can be smooth.' Only the weakest jumps (Fr₁ up to ≈1.7) are undular, riding a train of standing waves; above that they become breaking jumps with a rolling turbulent 'roller' and entrained air.

The standard classification by upstream Froude number (Chow, 1959):

• Fr₁ = 1–1.7: undular jump — gentle surface waves, little loss.
• Fr₁ = 1.7–2.5: weak jump — smooth roller, ~5–15% energy loss.
• Fr₁ = 2.5–4.5: oscillating jump — unsteady, sends waves downstream (avoid in design).
• Fr₁ = 4.5–9: steady jump — well-balanced, 45–70% loss, the engineer's favourite.
• Fr₁ > 9: strong jump — rough, up to 85% loss, heavy air entrainment.

One deep subtlety: the jump is where the specific-energy curve is double-valued. For a given discharge, each energy level above the minimum corresponds to two depths — one supercritical, one subcritical. The jump lets the flow hop from the low branch to the high branch while conserving momentum, paying the energy difference as the price of admission.

Supercritical vs. subcritical open-channel flow — the two states the jump connects

Property	Supercritical (upstream)	Subcritical (downstream)
Froude number Fr = U/√(gh)	Fr₁ > 1 (e.g. 3.5)	Fr₂ < 1 (e.g. 0.30)
Depth	Shallow, h₁ ≈ 0.3 mm (sink)	Deep, h₂ ≈ 3 mm
Speed	Fast, U₁ ≈ 1 m/s	Slow, U₂ ≈ 0.1 m/s
Wave behaviour	Flow outruns surface waves; disturbances swept downstream	Waves outrun flow; can travel upstream
Energy	High kinetic, low potential (lower branch of specific energy)	Lower kinetic, higher potential; ΔE dissipated in the jump
Analogy	Supersonic (Mach > 1)	Subsonic (Mach < 1)