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Physics calculator

Trapezoidal Normal Depth Calculator

Compute trapezoid uniform-flow yn from SI Manning with A=(b+z y)y and P=b+2 y √(1+z²). Runs locally. Not rectangular yn, not trapezoid yc, not Manning A,R, 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 · Trapezoid SI Manning yn invert with A=(b+z y)y and P=b+2y√(1+z²); + O3 mpmath tabulated yn.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Trapezoid SI Manning yn invert with A=(b+z y)y and P=b+2y√(1+z²); + O3 mpmath tabulated yn.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (depth, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not rectangular yn, not yc, not Manning A,R, not Chezy, and not Q=Av.
Numerical scope
O2: trapezoid SI Manning yn invert A=(b+z y)y, P=b+2y√(1+z²) vs a separate-module identity (≤2 ULP). Not rectangular yn, not yc, not Manning A,R, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated trapezoid-normal-depth vectors. Not rectangular yn, not yc, and not Manning A,R.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Trapezoidal uniform-flow normal depth from Q, n, S, b, and z. Invert Q.
Scope
Trapezoid uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=(b+z yn) yn; P=b+2 yn √(1+z²); SI k=1; z>0 is run/rise; yn is unknown in depth mode
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Trapezoid SI Manning yn invert with A=(b+z y)y and P=b+2y√(1+z²); + O3 mpmath tabulated yn.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3.cov
Verification revision
2026-09-26.o2-o3.cov · 1/1 property · digest f1a153685c1e
Legacy regression
6/6 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 — Normal depth in a trapezoidal section
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.

ManningQ = (1/n) A R^{2/3} S^{1/2}
AreaA = (b + z y) y
Wetted perimeterP = b + 2 y √(1+z²)
iSI k=1. z is run/rise, z>0. Geometry is bottom width b and side slope z, not A and R as inputs. yn/b∈[0.01, 10], n∈[0.008, 0.15], S∈[1e-6, 0.1], z∈(0, 10]. z=0 is the rectangular page. 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, b, and z

Discharge, Manning n, bed slope, bottom width, and side slope z (run/rise). Invert on the Q tab when yn is known. z=0 is the rectangular page.

2

Read yn

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

Example calculations

Common configurations with formula and result.

ϟ

b=2 m, z=1, n=0.03, S=0.001, Q≈2.30256 m³/s

Trapezoidal channel

A=(b+z yn) yn, P=b+2 yn √(1+z²), Q=(1/n) A R^{2/3} S^{1/2}
yn = 1.0000 m
ϟ

Invert Q

yn = 1 m, same n, S, b, z

Q from trapezoid Manning
Q ≈ 2.30256 m³/s

Trapezoidal Normal Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Trapezoid uniform flow: Q = (1/n) A R^{2/3} S^{1/2} with A = (b + z y) y and P = b + 2 y √(1+z²). This page solves yn. Rectangular yn and Manning with given A and R are different pages.
What it calculates
Trapezoidal uniform-flow normal depth from Q, n, S, b, and z. Invert Q.
Inputs
  • mode?
  • b
  • z
  • n
  • S
  • Q|yn
Outputs
  • yn
  • Q
  • b
  • z
  • n
  • S
  • A
  • P
  • R
  • v
  • mode
  • model
Formula
Q = (1/n) A R^{2/3} S^{1/2}; A = (b + z yn) yn; P = b + 2 yn √(1+z²)
Assumptions
  • Trapezoid uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=(b+z yn) yn; P=b+2 yn √(1+z²); SI k=1; z>0 is run/rise; yn is unknown in depth mode
  • yn/b∈[0.01, 10]; n∈[0.008, 0.15]; S∈[1e-6, 0.1]; z∈(0, 10]; geometry is b and z, not A and R
  • Not rectangular yn; not trapezoid yc; not Manning Q with given A,R; not Chezy C; not Q=Av
Units
  • SI; yn, b in m; Q in m³/s; S dimensionless; z dimensionless; n SI
Boundary conditions
  • missing b, z, n, S, Q, or yn when required → MISSING_REQUIRED_INPUT
  • yn/b, n, S, or z outside envelope → VALUE_OUT_OF_RANGE
  • z=0, A/R, yc/q, E, Chezy C, v, B, theta, g/Fr → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
Q=2.30256 b=2 z=1 n=0.03 S=0.001 → yn=1.0000
Validation cases

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

  • Q=2.30256 b=2 z=1 n=0.03 S=0.001 → yn ≈ 1.0000
  • mode=discharge yn=1 b=2 z=1 n=0.03 S=0.001 → Q ≈ 2.30256
  • Q=2.30256 b=2 z=0 n=0.03 S=0.001 → MIXED_INPUT_ENCODING
  • Q=2.30256 b=2 z=1 n=0.03 S=0.001 A=3 R=0.62 → 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 a trapezoidal section
    Supports: Uniform-flow yn from SI Manning with trapezoid A=(b+z y)y and P=b+2 y √(1+z²). Rectangular A=b yn is a different page. Not Manning with given A,R as a product page, not yc, not Chezy C
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute trapezoidal uniform-flow normal depth yn from Q, n, S, bottom width b, and side slope z.

Supported and not supported

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

Not supported — rectangular yn; trapezoid yc; Manning A,R; Chezy C; Q=Av; z=0; US k=1.486

Agent / API notes

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

{ "Q": 2.30256, "b": 2, "z": 1, "n": 0.03, "S": 0.001 }

REST and MCP call the same trapezoidNormalDepth wrapper as the page.

Calculator URL stays /calc/physics/trapezoidal-normal-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 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 rectangular normal depth?

No. Rectangular yn from A=b yn is /calc/physics/normal-depth. This page needs z>0. Sending z to the rectangular page, or z=0 here, is MIXED_INPUT_ENCODING.

Is this the Manning formula page?

No. /calc/physics/manning-formula takes A and R as inputs. This page takes b and z and solves yn. Sending A or R is MIXED_INPUT_ENCODING.

Is this trapezoidal critical depth?

No. Trapezoid Fr=1 from Q, b, z is /calc/physics/trapezoidal-critical-depth. Sending yc, q, or g 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 trapezoidNormalDepth wrapper.