HomeCalculatorsPhysicsFluid MechanicsHydraulic Jump Calculator
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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular sequent depth y2/y1 = ½(−1+√(1+8 Fr₁²)) with upstream and energy solves; + O3 mpmath tabulated depths.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Rectangular sequent depth y2/y1 = ½(−1+√(1+8 Fr₁²)) with upstream and energy solves; + O3 mpmath tabulated depths.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (sequent, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not trapezoidal jump, not Manning, not specific-energy alternate depth alone, and not Q=Av.
Numerical scope
O2: sequent depth y2/y1 = ½(−1+√(1+8 Fr₁²)) vs a separate-module identity (≤2 ULP). Not trapezoid jump, not specific energy alone, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated hydraulic-jump vectors. Not trapezoid jump, not Manning, and not Q=Av.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
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 · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular sequent depth y2/y1 = ½(−1+√(1+8 Fr₁²)) with upstream and energy solves; + O3 mpmath tabulated depths.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision
2026-09-26.o2-o3 · 1/1 property · digest 929f014e301e
Legacy regression
17/17 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 10/10 CVP boundary · 10/10 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-11 — Characteristic numbers — Froude number Fr
  • Chow, V.T. Open-Channel Hydraulics — Hydraulic jump
Sources
Evidence
2 legacy golden · 10 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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

1

Enter y₁

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

2

Enter Fr₁

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

3

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.

ϟ

Fr₁=2, y₁=1 m

Default g=9.80665 m/s²

y₂/y₁ = (√33 − 1)/2
y₂ ≈ 2.372 m
ϟ

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

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, M → 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 5/5 · 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, trapezoid z (that is /calc/physics/trapezoidal-hydraulic-jump)

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: Alternate Depth Calculator, Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Concentric Reducer Calculator, Critical Depth Calculator . Explore all Fluid Mechanics.

→ Alternate Depth Calculator Compute both rectangular depths at the same specific energy from E, Q, and width b. Runs locally. Not specific energy from y+v; not specific force M; not hydraulic jump y2; not critical depth yc as the product. › → Bernoulli Equation Calculator Incompressible Bernoulli along a streamline: solve P2, v2, or h2. Runs locally in your browser. › → Broad-Crested Weir Calculator Compute rectangular broad-crested weir discharge Q = C_d (2/3)^{3/2} b √g H^{3/2}. Default C_d=0.85. Runs locally. Not a thin-plate weir, not Manning, not Q=Av, and not an orifice. › → Cavitation Number Calculator Compute the cavitation number σ = (p − p_v) / (½ ρ v²). Runs locally. Not NPSH, not Thoma NPSH/H, not Weber, and not Euler Eu=Δp/(ρ v²). › → Chezy Formula Calculator Compute open-channel mean velocity V = C √(R S). Runs locally. Not Manning n, not Q=Av, not Hazen–Williams, and not Darcy. › → Colebrook Friction Factor Calculator Solve the Colebrook–White equation for Darcy friction factor f from Re and ε/D. Turbulent pipes Re≥4000. Runs locally. Not laminar 64/Re and not a Moody-chart lookup. › → Concentric Reducer Calculator Compute concentric-reducer loss h = K v₂² / (2g) with K = 0.5 (1 − A₂/A₁) referenced to the downstream speed. Runs locally. Not sudden expansion, not gradual contraction α, not vena-contracta C_c, not catalog K, and not Q=Av. › → Critical Depth Calculator Compute rectangular critical depth yc = (q²/g)^{1/3}. Runs locally. Not a Froude wrap, not specific energy, not a hydraulic jump, and not Q=Av. ›

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 specific force?

No. M = Q²/(gA) + A ȳ is /calc/physics/specific-force. This page takes Fr₁>1 and rejects Q. Sending M 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.

Is this a trapezoidal hydraulic jump?

No. Trapezoid sequent depths from specific force on A=(b+z y)y are /calc/physics/trapezoidal-hydraulic-jump. Sending z here is MIXED_INPUT_ENCODING.

Where does this run?

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