HomeCalculatorsPhysicsFluid MechanicsCritical Depth Calculator
Physics calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular yc = (q²/g)^{1/3} with unit-discharge invert; + O3 mpmath tabulated yc.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Rectangular yc = (q²/g)^{1/3} with unit-discharge invert; + O3 mpmath tabulated yc.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (depth, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not trapezoid critical depth, not alternate depth, not Manning yn, and not Q=Av alone.
Numerical scope
O2: yc=(q²/g)^{1/3} vs a separate-module identity (≤2 ULP). Not trapezoid critical depth, not alternate depth, not Manning, and not Q=Av alone. ≤2 ULP vs O3 applies only to the published tabulated critical-depth vectors. Not trapezoid yc, not Manning, and not alternate depth.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Rectangular critical depth from unit discharge. Invert q. REST accepts Q and b as q=Q/b.
Scope
Rectangular critical depth: yc = (q²/g)^{1/3}; q = Q/b; default g=9.80665 m/s²
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular yc = (q²/g)^{1/3} with unit-discharge invert; + O3 mpmath tabulated yc.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3.iface
Verification revision
2026-09-26.o2-o3.iface · 1/1 property · digest 5c7c205b89d1
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 · 11/11 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 5/5 CVP contract · Manifest
Sources
  • ISO 80000-4 — Mechanics — Length, speed, acceleration
  • Chow, V.T. Open-Channel Hydraulics — Critical flow
Sources
Evidence
2 legacy golden · 10 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 11/11 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.

Critical depthyc = (q² / g)^{1/3}
Unit discharge invertq = √(g yc³)
From dischargeq = Q / b
iSI rectangular prismatic channel. vc = √(g yc) is reported. Fr=1 is the definition of critical flow, not a Froude-number wrap. Not specific energy E, not a hydraulic jump, not Manning, not Q=Av.

How to use

1

Enter q

Unit discharge in m²/s. REST may send q or Q+b; the query layer uses q_m2_s.

2

Read yc

yc = (q²/g)^{1/3}. Invert q on the q tab when yc is known.

3

Optional g

Gravity defaults to 9.80665 m/s².

Example calculations

Common configurations with formula and result.

ϟ

q=1 m²/s

Default g=9.80665 m/s²

yc = (1 / 9.80665)^{1/3}
yc ≈ 0.46719 m
ϟ

Invert q

yc ≈ 0.46719 m

q = √(g yc³)
q = 1 m²/s

Critical Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Rectangular critical depth: yc = (q²/g)^{1/3} with q = Q/b and default g=9.80665 m/s². This page computes yc, not Fr = v/√(gL) and not E = y+v²/(2g).
What it calculates
Rectangular critical depth from unit discharge. Invert q. REST accepts Q and b as q=Q/b.
Inputs
  • mode?
  • q|yc
  • Q?
  • b?
  • g?
Outputs
  • yc
  • q
  • g
  • vc
  • mode
  • model
Formula
yc = (q² / g)^{1/3}
Assumptions
  • Rectangular critical depth: yc = (q²/g)^{1/3}; q = Q/b; default g=9.80665 m/s²
  • Critical speed vc = √(g yc) is reported, not an input; Fr=1 is the definition, not a Froude wrap
  • Not a Froude wrap, not specific energy E, not a hydraulic jump, not Manning, not Q=Av
Units
  • SI; yc in m; q in m²/s; Q in m³/s; b in m; g in m/s²
Boundary conditions
  • missing q or yc when required → MISSING_REQUIRED_INPUT
  • q, yc, Q, b, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
  • Fr/L, E/y, y1/y2, n, A, v, z → MIXED_INPUT_ENCODING
  • froude, energy, or jump as mode → INVALID_MODE
Example
q=1 → yc=(1/g)^{1/3}
Validation cases

3 published on this page · 17/17 tests · Production surface contract 5/5 · View evidence

  • q=1 → yc=(1/9.80665)^{1/3}
  • mode=unit_discharge yc=(1/g)^{1/3} → q=1
  • q=1 Fr=1 → MIXED_INPUT_ENCODING
Sources
  • ISO 80000-4 — Mechanics — Length, speed, acceleration
    Supports: yc is a length; q is discharge per width; g is acceleration
  • Chow, V.T. Open-Channel Hydraulics — Critical flow
    Supports: yc = (q²/g)^{1/3} for a rectangular channel at Fr=1
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute rectangular critical depth.

Supported and not supported

Supported — yc from q or Q+b · invert q · API physics.fluid.critical_depth

Not supported — Froude-only Fr=v/√(gL), specific energy E=y+v²/(2g), hydraulic jump sequent depths, Manning n, Q=Av

Agent / API notes

Capability id: physics.fluid.critical_depth · tool id: critical-depth · pin 1.0.0.

{ "q": 1 }

Modes: depth (default), unit_discharge. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, MIXED_INPUT_ENCODING, INVALID_MODE.

Calculator URL stays /calc/physics/critical-depth. 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 Slope 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 Slope Calculator Compute rectangular critical bed slope Sc such that yn=yc using SI Manning at y=yc. Runs locally. Not Manning A,R; not normal-depth yn; not critical-depth yc as the product; not Chezy; 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 returns yc. Sending Fr or L is MIXED_INPUT_ENCODING.

Is this trapezoidal critical depth?

No. Fr=1 on a trapezoid from Q, b, and z is /calc/physics/trapezoidal-critical-depth. Sending z is MIXED_INPUT_ENCODING.

Is this specific energy?

No. E = y + v²/(2g) is /calc/physics/specific-energy. Sending E or y is MIXED_INPUT_ENCODING.

Is this a hydraulic jump?

No. Sequent depths are /calc/physics/hydraulic-jump. Sending y1 or y2 is MIXED_INPUT_ENCODING.

Is this Q = A·v?

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

Where does this run?

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