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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 Engine tested · Source checked · 6/6 tests · Production surface contract 1/1 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
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 · 6/6 tests · Production surface contract 1/1 · Source checked · v1.0.0
Named expert review
Optional · Not performed
Sources
  • ISO 80000-4 — Mechanics — Volume flow rate and hydraulic radius
  • Chow, V.T. Open-Channel Hydraulics — Critical slope and critical depth
Sources
Evidence
6 property · Production surface contract 1/1 · 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; 164 CURRENT) @ 2026-09-18T21:41:56.329Z
Semantic contract
PASS

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 · 6/6 tests · Production surface contract 1/1 · 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: Normal depth, Critical depth, Manning formula, Specific energy, Chezy formula .

yn Normal depth › yc Critical depth › Q Manning formula › E Specific energy › C Chezy formula ›
}

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.