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Trapezoidal Alternate Depth Calculator

Compute both trapezoid depths at the same specific energy from E, Q, b, and z with A=(b+z y)y. Runs locally. Not rectangular alternate depth; not trapezoid E; not trapezoid yc; not trapezoid yn.

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
Both trapezoid depths at the same E from E, Q, b, and z. Invert E from one y.
Scope
Trapezoid alternate depths: E = y + Q²/(2g A²) with A=(b+z y)y; both y_sub>yc and y_super<yc at the same E, Q, b, z; SI; default g=9.80665; z>0 is run/rise
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 — Energy and volume flow
  • Chow, V.T. Open-Channel Hydraulics — Specific energy and alternate depths in a trapezoidal section
Sources
Evidence
5 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @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.

Specific energyE = y + Q² / (2g A²)
SectionA = (b + z y) y
CriticalEc = yc + D_c / 2
iSI trapezoidal prismatic channel. z is run/rise, z>0. Default g=9.80665 m/s². Require E > Ec. 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, yc, or yn as inputs.

How to use

1

Enter E, Q, b, and z

Specific energy, discharge, bottom width, and side slope z (run/rise). z=0 is the rectangular page. 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=1.02266 m, Q=2 m³/s, b=2 m, z=1

Trapezoidal channel, default g=9.80665

E = y + Q²/(2g A²)
y_sub ≈ 1.0000 m · y_super ≈ 0.22735 m
ϟ

Invert E

y = 1 m, same Q, b, z

E = y + Q²/(2g A²)
E ≈ 1.02266 m

Trapezoidal Alternate Depth calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Trapezoid alternate depths: E = y + Q²/(2g A²) with A=(b+z y)y. The same E, Q, b, and z give y_sub > yc and y_super < yc. Rectangular both y from E, Q, b is a different page. Trapezoid E from y and Q is a different page.
What it calculates
Both trapezoid depths at the same E from E, Q, b, and z. Invert E from one y.
Inputs
  • mode?
  • b
  • z
  • Q
  • E|y
  • g?
Outputs
  • E
  • Q
  • b
  • z
  • 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+z y)y
Assumptions
  • Trapezoid alternate depths: E = y + Q²/(2g A²) with A=(b+z y)y; both y_sub>yc and y_super<yc at the same E, Q, b, z; SI; default g=9.80665; z>0 is run/rise
  • Unknown is the pair (depths mode) or E (energy invert from one y); E>Ec=yc+D_c/2; Fr is reported not an input
  • Not rectangular alternate depth; not trapezoid E from y+Q; not trapezoid yc as the product; not trapezoid yn; not Manning n; not Q=Av
Units
  • SI; y, b, E in m; Q in m³/s; z dimensionless
Boundary conditions
  • missing b, z, Q, E, or y when required → MISSING_REQUIRED_INPUT
  • E ≤ Ec or y/b outside [0.01, 20] or z>10 → VALUE_OUT_OF_RANGE
  • z=0, A/v, y1/y2, M, Fr/L, n/C/S/R/Sc, yn/yc/q, H/Cd → MIXED_INPUT_ENCODING
  • unknown mode → INVALID_MODE
Example
E=1.02266 Q=2 b=2 z=1 → y_sub≈1.0000
Validation cases

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

  • E=1.02266 Q=2 b=2 z=1 → y_sub ≈ 1.0000 m
  • mode=energy y=1 Q=2 b=2 z=1 → E ≈ 1.02266 m
  • E=1.02266 Q=2 b=2 z=0 → MIXED_INPUT_ENCODING
  • E=1.02266 Q=2 b=2 z=1 v=1 → 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 in a trapezoidal section
    Supports: The same E gives two depths when E>Ec. Rectangular both y from E, Q, b is a different product. Trapezoid E from y and Q is a different product.
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute both trapezoid depths at the same specific energy from E, Q, bottom width b, and side slope z.

Supported and not supported

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

Not supported — rectangular alternate depth; trapezoid E from y+Q; trapezoid yc as the product; trapezoid yn; Manning n; Q=Av; z=0

Agent / API notes

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

{ "E": 1.02266, "Q": 2, "b": 2, "z": 1 }

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

Calculator URL stays /calc/physics/trapezoidal-alternate-depth. There is no /calc/fluid.

Other calculators in this family: Alternate depth, Trapezoidal specific energy, Trapezoidal critical depth, Trapezoidal normal depth, Specific energy, Froude number, Trapezoidal hydraulic jump .

y Alternate depth › E Trapezoidal specific energy › yc Trapezoidal critical depth › yn Trapezoidal normal depth › E Specific energy › Fr Froude number › y₂ Trapezoidal hydraulic jump ›
}

Frequently asked questions

Key distinctions behind the calculation.

Is this rectangular alternate depth?

No. Both rectangular y from E, Q, and b are /calc/physics/alternate-depth. This page needs z>0. Sending z to the rectangular page, or z=0 here, is MIXED_INPUT_ENCODING.

Is this trapezoidal specific energy?

No. E = y + Q²/(2g A²) from y, Q, b, z is /calc/physics/trapezoidal-specific-energy. That page does not invert y. Sending mode=energy|discharge as if this were E, or sending v, is MIXED_INPUT_ENCODING.

Is this trapezoidal critical depth?

No. Trapezoid Fr=1 from Q, b, z is /calc/physics/trapezoidal-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 trapezoidal normal depth or Q = A·v?

No. Trapezoid yn from Manning is /calc/physics/trapezoidal-normal-depth. Volumetric flow from a given area and velocity is /calc/physics/flow-rate. Sending n, S, yn, A, or v is MIXED_INPUT_ENCODING.

Is this a trapezoidal hydraulic jump?

No. Trapezoid sequent depths from specific force are /calc/physics/trapezoidal-hydraulic-jump. Sending y1, y2, or mode=sequent is MIXED_INPUT_ENCODING.

Where does this run?

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