HomeCalculatorsPhysicsFluid MechanicsCritical Slope Calculator
Physics calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular Sc at yn=yc from SI Manning; + O3 mpmath tabulated Sc.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Rectangular Sc at yn=yc from SI Manning; + O3 mpmath tabulated Sc.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (slope, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not yn from given S, not trapezoid, not Manning A,R, not Chezy, and not Q=Av.
Numerical scope
O2: rectangular Sc at yn=yc from SI Manning vs a separate-module identity (≤2 ULP). Not normal-depth yn from given S, not trapezoid, not Manning A,R, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated critical-slope vectors. Not yn from given S, not trapezoid, and not Manning A,R.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Rectangular critical bed slope Sc from Q, n, and width b so that yn=yc. Invert Q.
Scope
Rectangular yn=yc: Sc=[n Q/(A R^{2/3})]^2 at yc=(q²/g)^{1/3}; A=b yc; R=b yc/(b+2 yc); SI k=1; Sc is unknown in slope mode
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular Sc at yn=yc from SI Manning; + O3 mpmath tabulated Sc.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3.cov.iut
Verification revision
2026-09-26.o2-o3.cov.iut · 1/1 property · digest 81c6323995ec
Legacy regression
7/7 tests · Production surface contract 6/6
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
5/5 golden · 1/1 CVP boundary · 2/2 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
  • ISO 80000-4 — Mechanics — Volume flow rate and hydraulic radius
  • Chow, V.T. Open-Channel Hydraulics — Critical slope and critical depth
Sources
Evidence
1 legacy golden · legacy regression suite · 5/5 oracle-backed golden · 2/2 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}
Manning at ycSc = [n Q / (A R^{2/3})]^2
SectionA = b yc, R = b yc / (b + 2 yc)
iSI k=1. yc/b∈[0.2, 10] so Sc(yc) is one-to-one on this seed. n∈[0.008, 0.15], Sc∈[1e-6, 0.1]. Default g=9.80665. Do not send query v — that is calculation_version. Mean speed is result v. Do not send bed slope S — that is normal depth.

How to use

1

Enter Q, n, and b

Discharge, Manning n, and rectangular width. Invert on the Q tab when Sc is known.

2

Read Sc

Sc is the bed slope at which yn=yc. yc, A, R, and Fr=1 are reported.

Example calculations

Common configurations with formula and result.

ϟ

b=2 m, n=0.03, Q≈3.4546 m³/s

Rectangular channel

yc=(q²/g)^{1/3}, Sc=[n Q/(A R^{2/3})]^2
Sc = 0.02000
ϟ

Invert Q

Sc = 0.02, same n, b

Q from rectangular Manning at yc
Q ≈ 3.45460 m³/s

Critical Slope calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Rectangular critical slope: Sc such that yn=yc. yc=(q²/g)^{1/3}. Sc=[n Q/(A R^{2/3})]^2 with A=b yc and R=b yc/(b+2 yc). Normal depth from a given S is a different page. Critical depth yc from q is a different page.
What it calculates
Rectangular critical bed slope Sc from Q, n, and width b so that yn=yc. Invert Q.
Inputs
  • mode?
  • b
  • n
  • Q|Sc
  • g?
Outputs
  • Sc
  • Q
  • yc
  • b
  • n
  • g
  • A
  • P
  • R
  • v
  • Fr
  • mode
  • model
Formula
yc=(q²/g)^{1/3}; Sc=[n Q/(A R^{2/3})]^2; A=b yc; R=b yc/(b+2 yc)
Assumptions
  • Rectangular yn=yc: Sc=[n Q/(A R^{2/3})]^2 at yc=(q²/g)^{1/3}; A=b yc; R=b yc/(b+2 yc); SI k=1; Sc is unknown in slope mode
  • yc/b∈[0.2, 10]; n∈[0.008, 0.15]; Sc∈[1e-6, 0.1]; geometry is b, not A and R; default g=9.80665
  • Not Manning Q with given A,R,S; not normal-depth yn invert; not critical-depth yc as the product; not Chezy C; not Q=Av; not trapezoid z
Units
  • SI; yc, b in m; Q in m³/s; Sc dimensionless; n SI
Boundary conditions
  • missing b, n, Q, or Sc when required → MISSING_REQUIRED_INPUT
  • yc/b, n, or Sc outside envelope → VALUE_OUT_OF_RANGE
  • A/R, yn, yc/q, S, E, Chezy C, v, B, z, y1, H, Fr → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
Q=3.4546 b=2 n=0.03 → Sc=0.02000
Validation cases

4 published on this page · 7/7 tests · Production surface contract 6/6 · View evidence

  • Q=3.4546 b=2 n=0.03 → Sc ≈ 0.02000
  • mode=discharge Sc=0.02 b=2 n=0.03 → Q ≈ 3.45460
  • Q=3.4546 b=2 n=0.03 A=1 R=0.3 → MIXED_INPUT_ENCODING
  • Q=3.4546 b=2 n=0.03 yn=1 → MIXED_INPUT_ENCODING
Sources
  • ISO 80000-4 — Mechanics — Volume flow rate and hydraulic radius
    Supports: Q is volume flow; hydraulic radius R = A/P
  • Chow, V.T. Open-Channel Hydraulics — Critical slope and critical depth
    Supports: Sc is the bed slope at which uniform yn equals critical yc. Not Manning with given A,R as a product page, not yc as the unknown, not Chezy C
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute rectangular critical bed slope Sc from Q, n, and width b so that yn = yc.

Supported and not supported

Supported — Sc from Q, n, b · invert Q · API physics.fluid.critical_slope

Not supported — Manning A,R; normal-depth yn; critical-depth yc as the product; specific energy E; Chezy C; Q=Av; trapezoid z; US k=1.486

Agent / API notes

Capability id: physics.fluid.critical_slope · tool id: critical-slope · pin 1.0.0.

{ "Q": 3.4546, "b": 2, "n": 0.03 }

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 Manning formula page?

No. /calc/physics/manning-formula takes A and R as inputs. This page takes rectangular width b and Q (or Sc). Sending A, R, or Sc to Manning is MIXED_INPUT_ENCODING.

Is this normal depth?

No. yn from a given bed slope S is /calc/physics/normal-depth. Sending S or yn is MIXED_INPUT_ENCODING.

Is this critical depth?

No. yc = (q²/g)^{1/3} is /calc/physics/critical-depth. This page reports yc at Fr=1 but the unknown is Sc. Sending yc, q, or Sc to critical-depth is MIXED_INPUT_ENCODING.

Is this Chezy?

No. V = C √(R S) is /calc/physics/chezy-formula. Sending Chezy C or Sc to Chezy is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. Sending v is MIXED_INPUT_ENCODING. Result v = Q/A is reported.

Where does this run?

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