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Physics calculator

Normal Depth Calculator

Compute rectangular uniform-flow normal depth yn from SI Manning with A=b yn and R=b yn/(b+2 yn). Runs locally. Not Manning A,R; not critical depth; not specific energy; not Chezy; not Q=Av.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 5/5 tests · Production surface contract 1/1 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Rectangular uniform-flow normal depth from Q, n, S, and width b. Invert Q.
Scope
Rectangular uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=b yn; R=b yn/(b+2 yn); SI k=1; yn is unknown in depth mode
Verification
Engine tested · 5/5 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 — Normal depth in uniform flow
Sources
Evidence
5 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) @ 2026-09-18T13:56:15.885Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

ManningQ = (1/n) A R^{2/3} S^{1/2}
AreaA = b yn
Hydraulic radiusR = b yn / (b + 2 yn)
iSI k=1. Geometry is rectangular width b, not A and R as inputs. yn/b∈[0.01, 10], n∈[0.008, 0.15], S∈[1e-6, 0.1]. Not US k=1.486. Do not send query v — that is calculation_version. Mean speed is result v.

How to use

1

Enter Q, n, S, and b

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

2

Read yn

yn is the uniform-flow depth. A, R, and v=Q/A are reported.

Example calculations

Common configurations with formula and result.

ϟ

b=2 m, n=0.03, S=0.001, Q≈1.32807 m³/s

Rectangular channel

A=b yn, R=A/(b+2 yn), Q=(1/n) A R^{2/3} S^{1/2}
yn = 1.0000 m
ϟ

Invert Q

yn = 1 m, same n, S, b

Q from rectangular Manning
Q ≈ 1.32807 m³/s

Normal Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Rectangular uniform flow: Q = (1/n) A R^{2/3} S^{1/2} with A = b yn and R = b yn / (b + 2 yn). This page solves yn. Manning with given A and R is a different page.
What it calculates
Rectangular uniform-flow normal depth from Q, n, S, and width b. Invert Q.
Inputs
  • mode?
  • b
  • n
  • S
  • Q|yn
Outputs
  • yn
  • Q
  • b
  • n
  • S
  • A
  • P
  • R
  • v
  • mode
  • model
Formula
Q = (1/n) A R^{2/3} S^{1/2}; A = b yn; R = b yn / (b + 2 yn)
Assumptions
  • Rectangular uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=b yn; R=b yn/(b+2 yn); SI k=1; yn is unknown in depth mode
  • yn/b∈[0.01, 10]; n∈[0.008, 0.15]; S∈[1e-6, 0.1]; geometry is b, not A and R
  • Not Manning Q with given A,R; not critical depth yc=(q²/g)^{1/3}; not specific energy E; not Chezy C; not Q=Av
Units
  • SI; yn, b in m; Q in m³/s; S dimensionless; n SI
Boundary conditions
  • missing b, n, S, Q, or yn when required → MISSING_REQUIRED_INPUT
  • yn/b, n, or S outside envelope → VALUE_OUT_OF_RANGE
  • A/R, yc/q, E, Chezy C, v, B, z, y1, H → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
Q=1.32807 b=2 n=0.03 S=0.001 → yn=1.0000
Validation cases

4 published on this page · 5/5 tests · Production surface contract 1/1 · View evidence

  • Q=1.32807 b=2 n=0.03 S=0.001 → yn ≈ 1.0000
  • mode=discharge yn=1 b=2 n=0.03 S=0.001 → Q ≈ 1.32807
  • Q=1.32807 b=2 n=0.03 S=0.001 A=2 R=0.5 → MIXED_INPUT_ENCODING
  • Q=1.32807 b=2 n=0.03 S=0.001 yc=0.3 → 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 — Normal depth in uniform flow
    Supports: Uniform-flow yn from SI Manning with rectangular A=b yn and R=A/(b+2 yn). Not Manning with given A,R as a product page, not yc, not E, not Chezy C
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute rectangular uniform-flow normal depth yn from Q, n, S, and width b.

Supported and not supported

Supported — yn from Q, n, S, b · invert Q · API physics.fluid.normal_depth

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

Agent / API notes

Capability id: physics.fluid.normal_depth · tool id: normal-depth · pin 1.0.0.

{ "Q": 1.32807, "b": 2, "n": 0.03, "S": 0.001 }

Other calculators in this family: Manning formula, Critical depth, Specific energy, Chezy formula, Hydraulic jump .

Q Manning formula › yc Critical depth › E Specific energy › C Chezy formula › y₂ Hydraulic jump ›
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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 and returns Q, v, n, or S. This page takes rectangular width b and solves yn. Sending A or R is MIXED_INPUT_ENCODING.

Is this critical depth?

No. yc = (q²/g)^{1/3} is /calc/physics/critical-depth. Sending yc or q is MIXED_INPUT_ENCODING.

Is this specific energy?

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

Is this Chezy?

No. V = C √(R S) is /calc/physics/chezy-formula. Sending Chezy C 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 normalDepth wrapper.