Physics calculator

Hydraulic Jump Calculator

Compute rectangular sequent depths y₂/y₁ = (√(1+8 Fr₁²) − 1)/2. Fr₁ > 1. Runs locally. Not a Froude wrap, not Manning, not a weir, and not Q=Av.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary Engine tested · Source checked · 17/17 tests · Production surface contract 3/3 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
Model: Rectangular sequent depths and jump energy loss. Invert y₁. Optional ΔE from y₁ and y₂.
Scope: Rectangular hydraulic jump: y2/y1 = (√(1+8 Fr₁²) − 1)/2; Fr₁ > 1
Verification: Engine tested · 17/17 tests · Production surface contract 3/3 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: ISO 80000-11 — Characteristic numbers — Froude number Fr
Chow, V.T. Open-Channel Hydraulics — Hydraulic jump
Sources
Evidence: 2 golden · 10 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status PASS (0 stale; 163 CURRENT) @ 2026-09-17T09:03:11.906Z
Semantic contract: PASS

Formulas

Core equations used by this calculator.

Sequent depthy₂/y₁ = (√(1+8 Fr₁²) − 1)/2

Energy lossΔE = (y₂ − y₁)³ / (4 y₁ y₂)

Downstream FroudeFr₂ = Fr₁ / (y₂/y₁)^{3/2}

iSI rectangular prismatic channel. Default g=9.80665 m/s². This page computes conjugate depths, not Fr = v/√(gL) itself. Not Manning, not Chezy, not a weir, not Q=Av.

How to use

Enter y₁

Upstream depth in metres. Invert for y₁ on the y₁ tab.

Enter Fr₁

Upstream Froude number. Must be greater than 1. REST may send v instead of Fr.

Read y₂

y₂ = y₁ (√(1+8 Fr₁²) − 1)/2. ΔE and Fr₂ are reported with the result.

Example calculations

Common configurations with formula and result.

Fr₁=2, y₁=1 m

Default g=9.80665 m/s²

y₂/y₁ = (√33 − 1)/2

y₂ ≈ 2.372 m

From velocity

y₁=1 m, v=2√g

Fr₁ = v/√(g y₁) = 2

same y₂

Hydraulic Jump calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 17/17 tests · Production surface contract 3/3
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: Rectangular hydraulic jump: y₂/y₁ = (√(1+8 Fr₁²) − 1)/2 with Fr₁ > 1. Energy loss ΔE = (y₂ − y₁)³ / (4 y₁ y₂).
What it calculates: Rectangular sequent depths and jump energy loss. Invert y₁. Optional ΔE from y₁ and y₂.
Inputs: mode?
y1|y2
Fr|v
g?
Outputs: y1
y2
Fr1
Fr2
v1
v2
dE
g
mode
model
Formula: y₂/y₁ = (√(1+8 Fr₁²) − 1)/2
Assumptions: Rectangular hydraulic jump: y2/y1 = (√(1+8 Fr₁²) − 1)/2; Fr₁ > 1
ΔE = (y2 − y1)³ / (4 y1 y2); Fr₂ = Fr₁ / (y2/y1)^{3/2}
Not a Froude wrap, not Manning, not Chezy, not a weir, not Q=Av
Units: SI; y in m; v in m/s; g in m/s²; Fr dimensionless
Boundary conditions: missing y₁, Fr, or v when required → MISSING_REQUIRED_INPUT
Fr₁ ≤ 1 → VALUE_OUT_OF_RANGE
y, Fr, v, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
Re/ρ/μ, We/σ, n, A/Q, L, H/C_d → MIXED_INPUT_ENCODING
froude, manning, or weir as mode → INVALID_MODE
Example: y1=1 Fr=2 → y2=(√33−1)/2
Validation cases: 3 published on this page · 17/17 tests · Production surface contract 3/3 · View evidence
y1=1 Fr=2 → y2=(√33−1)/2
y1=1 v=2√g → same y2
y1=1 Fr=1 → VALUE_OUT_OF_RANGE
Sources: ISO 80000-11 — Characteristic numbers — Froude number Fr
Supports: Fr = v / √(l g) used as Fr₁ in the sequent-depth relation
Chow, V.T. Open-Channel Hydraulics — Hydraulic jump
Supports: Bélanger sequent depths and ΔE = (y2 − y1)³ / (4 y1 y2) for a rectangular jump
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute rectangular sequent depths for a hydraulic jump.

Supported and not supported

Supported — y₂ from y₁ and Fr₁ (or v) · invert y₁ · ΔE from y₁ and y₂ · API physics.fluid.hydraulic_jump

Not supported — Froude-only Fr=v/√(gL), Manning n, Chezy C, weir Q(H), Q=Av

Agent / API notes

Capability id: physics.fluid.hydraulic_jump · tool id: hydraulic-jump · pin 1.0.0.

{ "y1": 1, "Fr": 2 }

From velocity: { "y1": 1, "v": 6.26312 }. Modes: sequent (default), upstream, energy. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/hydraulic-jump. There is no /calc/fluid.

Other calculators in this family: Froude number, Manning formula, Chezy formula, Weir discharge .

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Frequently asked questions

Key distinctions behind the calculation.

Is this the Froude number?

No. Fr = v / √(g L) is /calc/physics/froude-number. This page takes Fr₁ (or v) and returns sequent depth. Sending L is MIXED_INPUT_ENCODING.

Is this Manning?

No. Uniform open-channel Q from n is /calc/physics/manning-formula. Sending n is MIXED_INPUT_ENCODING.

Is this a weir?

No. Thin-plate Q(H) is /calc/physics/weir-discharge. Sending H or C_d is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from area and velocity is /calc/physics/flow-rate. Sending A or Q is MIXED_INPUT_ENCODING.

Where does this run?

Locally in the browser by default. REST and MCP call the same fluid-engine hydraulicJump wrapper.