Physics calculator

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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Trapezoid alternate depths from E = y + Q²/(2gA²); + O3 mpmath tabulated depths.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: Trapezoid alternate depths from E = y + Q²/(2gA²); + O3 mpmath tabulated depths.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (depths, solve, alias, awkward, invalid-domain) · Matrix
Deferred: Not rectangular alternate depths, not jump y2, not M, and not Q=Av.
Numerical scope: O2: trapezoid alternate depths from E=y+Q²/(2gA²) vs a separate-module identity (≤2 ULP). Not rectangular alternate depths, not jump y2, not M, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated trapezoid-alternate-depth vectors. Not rectangular alternate depths, not jump y2, and not M.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
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 · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Trapezoid alternate depths from E = y + Q²/(2gA²); + 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 80baeec2ac43
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: 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: 1 legacy golden · legacy regression suite · 4/4 oracle-backed golden · 2/2 invalid · Artifact integrity PASS
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 + 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

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.

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.

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

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

Engine tested 6/6 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

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 · 6/6 tests · Production surface contract 6/6 · 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 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.

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.