HomeCalculatorsPhysicsFluid MechanicsAlternate Depth Calculator
Physics calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular alternate depths from E = y + Q²/(2gA²) with energy invert; + O3 mpmath tabulated depths.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
Rectangular alternate depths from E = y + Q²/(2gA²) with energy invert; + O3 mpmath tabulated depths.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (depths, solve, alias, awkward, invalid-domain) · Matrix
Deferred
Not jump y2, not specific force M, not trapezoid alternate depths, and not Q=Av.
Numerical scope
O2: rectangular alternate depths from E=y+Q²/(2gA²) vs a separate-module identity (≤2 ULP). Not jump y2, not specific force M, not trapezoid, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated alternate-depth vectors. Not jump y2, not M, and not trapezoid.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Both rectangular depths at the same E from E, Q, and b. Invert E from one y.
Scope
Rectangular alternate depths: E = y + Q²/(2g A²) with A=b y; both y_sub>yc and y_super<yc at the same E, Q, b; SI; default g=9.80665
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Rectangular alternate depths from E = y + Q²/(2gA²) with energy invert; + O3 mpmath tabulated depths.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-26.o2-o3
Verification revision
2026-09-26.o2-o3 · 1/1 property · digest b851963fc9ba
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
4/4 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
Sources
Evidence
1 legacy golden · legacy regression suite · 4/4 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.

Specific energyE = y + Q² / (2g A²)
SectionA = b y
CriticalEc = (3/2) yc
iSI rectangular prismatic channel. Default g=9.80665 m/s². Require E > Ec = 1.5 yc. Do not send query v — that is calculation_version. Mean speeds are result v_sub and v_super. Do not send M, y1, y2, Fr, A, or v as inputs.

How to use

1

Enter E, Q, and b

Specific energy, discharge, and rectangular width. Invert on the E tab when one y is known.

2

Read y_sub and y_super

Both roots at the same E. yc, Ec, and Fr on each branch are reported.

Example calculations

Common configurations with formula and result.

ϟ

E=1.05099 m, Q=2 m³/s, b=2 m

Rectangular channel, default g=9.80665

E = y + Q²/(2g A²)
y_sub ≈ 1.0000 m · y_super ≈ 0.25273 m
ϟ

Invert E

y = 1 m, same Q, b

E = y + Q²/(2g (b y)²)
E ≈ 1.05099 m

Alternate Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Rectangular alternate depths: E = y + Q²/(2g A²) with A = b y. The same E, Q, and b give y_sub > yc and y_super < yc. This page takes E, Q, and b. Specific energy E from y and v is a different page. Critical depth yc from q is a different page.
What it calculates
Both rectangular depths at the same E from E, Q, and b. Invert E from one y.
Inputs
  • mode?
  • b
  • Q
  • E|y
  • g?
Outputs
  • E
  • Q
  • b
  • y_sub
  • y_super
  • y
  • y_alt
  • yc
  • Ec
  • v_sub
  • v_super
  • Fr_sub
  • Fr_super
  • g
  • mode
  • model
Formula
E=y+Q²/(2gA²); A=b y
Assumptions
  • Rectangular alternate depths: E = y + Q²/(2g A²) with A=b y; both y_sub>yc and y_super<yc at the same E, Q, b; SI; default g=9.80665
  • Unknown is the pair (depths mode) or E (energy invert from one y); E>Ec=1.5 yc; Fr is reported not an input
  • Not specific energy E from y+v; not specific force M; not hydraulic jump y2 from Fr₁>1; not critical-depth yc as the product; not Manning n; not Q=Av
Units
  • SI; y, b, E in m; Q in m³/s
Boundary conditions
  • missing b, Q, E, or y when required → MISSING_REQUIRED_INPUT
  • E ≤ Ec or y/b outside [0.01, 10] → VALUE_OUT_OF_RANGE
  • A/v, y1/y2, M, Fr/L, n/C/S/R/Sc, yn/yc/q, z, H/Cd → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
E=1.05099 Q=2 b=2 → y_sub≈1.0000
Validation cases

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

  • E=1.05099 Q=2 b=2 → y_sub ≈ 1.0000 m
  • mode=energy y=1 Q=2 b=2 → E ≈ 1.05099 m
  • E=1.05099 Q=2 b=2 v=1 → MIXED_INPUT_ENCODING
  • Q=2 b=2 E=1.05099 on critical-depth → MIXED_INPUT_ENCODING
Sources
  • ISO 80000-4 — Mechanics — Energy and volume flow
    Supports: E and y are lengths; Q is volume flow
  • Chow, V.T. Open-Channel Hydraulics — Specific energy and alternate depths
    Supports: The same E gives two depths when E>Ec. E from y and v is a different product. yc from q is a different product.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute both rectangular depths at the same specific energy from E, Q, and width b.

Supported and not supported

Supported — y_sub and y_super from E, Q, b · invert E · API physics.fluid.alternate_depth

Not supported — specific energy from y+v; specific force M; hydraulic jump y2 from Fr₁>1; critical depth yc as the product; Manning n; Q=Av; trapezoid z (that is /calc/physics/trapezoidal-alternate-depth)

Agent / API notes

Capability id: physics.fluid.alternate_depth · tool id: alternate-depth · pin 1.0.0.

{ "E": 1.05099, "Q": 2, "b": 2 }

Other calculators in this family: Bernoulli Equation Calculator, Broad-Crested Weir Calculator, Cavitation Number Calculator, Chezy Formula Calculator, Colebrook Friction Factor Calculator, Concentric Reducer Calculator, Critical Depth Calculator, Critical Slope Calculator . Explore all Fluid Mechanics.

→ 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. › → 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 specific energy?

No. E = y + v²/(2g) from y and v is /calc/physics/specific-energy. That page rejects Q and b. Sending v is MIXED_INPUT_ENCODING.

Is this specific force?

No. M = Q²/(gA) + A ȳ is /calc/physics/specific-force. Sending M is MIXED_INPUT_ENCODING.

Is this critical depth?

No. yc = (q²/g)^{1/3} is /calc/physics/critical-depth. This page reports yc but the unknowns are the two y at E > Ec. Sending yc or q is MIXED_INPUT_ENCODING.

Is this a hydraulic jump or Q = A·v?

No. Sequent y2 from Fr₁>1 is /calc/physics/hydraulic-jump. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. Sending y1, y2, A, or v is MIXED_INPUT_ENCODING.

Is this trapezoidal alternate depth?

No. Both trapezoid y from E, Q, b, and z are /calc/physics/trapezoidal-alternate-depth. Sending z here is MIXED_INPUT_ENCODING.

Where does this run?

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