Physics calculator

Trapezoidal Specific Energy Calculator

Compute trapezoid specific energy E=y+Q²/(2gA²) with A=(b+z y)y. Runs locally. Not rectangular E=y+v²/(2g), not trapezoid yc, not trapezoid yn, not Manning A,R, 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 E = y + Q²/(2gA²) with A=(b+z y)y and discharge invert; + O3 mpmath tabulated E.

Input interpretation: Enter values to calculate.
Result: —
Verified scope: Trapezoid E = y + Q²/(2gA²) with A=(b+z y)y and discharge invert; + O3 mpmath tabulated E.
Assurance: Engineering
Declared partition coverage: PASS · 5/5 declared partitions (energy, solve, alias, awkward, invalid-domain) · Matrix
Deferred: Not rectangular E=y+v²/(2g), not yc, not yn, not alternate depths, and not Q=Av.
Numerical scope: O2: trapezoid E=y+Q²/(2gA²) with A=(b+z y)y vs a separate-module identity (≤2 ULP). Not rectangular E=y+v²/(2g), not yc, not yn, and not Q=Av. ≤2 ULP vs O3 applies only to the published tabulated trapezoid-specific-energy vectors. Not rectangular E, not yc, and not yn.
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Trapezoidal specific energy from y, Q, b, and z. Invert Q.
Scope: Trapezoid specific energy: E=y+Q²/(2g A²) with A=(b+z y)y; 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 E = y + Q²/(2gA²) with A=(b+z y)y and discharge invert; + O3 mpmath tabulated E.· 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 8fde62db5939
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, length, speed
Chow, V.T. Open-Channel Hydraulics — Specific energy 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)

AreaA = (b + z y) y

Discharge invertQ = A √(2g(E − y))

iSI trapezoidal prismatic channel. z is horizontal/vertical. v=Q/A is reported. Not rectangular E from y and v, not yc, not yn, not Manning, not Q=Av. Do not invert y here — that pair is alternate depth.

How to use

Enter y, Q, b, and z

Depth in m, discharge in m³/s, bottom width in m, and side slope z (run/rise). z=0 is the rectangular page. Invert Q on the Q tab when E is known.

Read E

E = y + Q²/(2g A²). Require E > y on the Q invert.

Optional g

Gravity defaults to 9.80665 m/s².

Example calculations

Common configurations with formula and result.

y=1 m, Q=2 m³/s, b=2 m, z=1

Default g=9.80665 m/s²

E = y + Q²/(2g A²)

E ≈ 1.02266 m

Invert Q

E ≈ 1.02266 m, y=1 m, b=2 m, z=1

Q = A √(2g(E − y))

Q ≈ 2 m³/s

Trapezoidal Specific Energy 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 specific energy: E = y + Q²/(2g A²) with A=(b+z y)y. Default g=9.80665 m/s². z is run/rise, z>0. Rectangular E=y+v²/(2g) is a different page.
What it calculates: Trapezoidal specific energy from y, Q, b, and z. Invert Q.
Inputs: mode?
y
b
z
Q|E
g?
Outputs: E
y
Q
b
z
A
T
v
Fr
g
mode
model
Formula: E = y + Q² / (2g A²)
Assumptions: Trapezoid specific energy: E=y+Q²/(2g A²) with A=(b+z y)y; SI; default g=9.80665; z>0 is run/rise
Unknown is E (energy mode) or Q (discharge invert); v=Q/A is reported; not y invert (that is alternate depth)
Not rectangular E=y+v²/(2g); not trapezoid yc; not trapezoid yn; not Manning A,R; not Q=Av
Units: SI; E and y in m; Q in m³/s; b in m; z dimensionless; g in m/s²
Boundary conditions: missing y, Q, b, z, or E when required → MISSING_REQUIRED_INPUT
y, Q, b, z, E, or g ≤ 0 → VALUE_MUST_BE_POSITIVE
E ≤ y → VALUE_OUT_OF_RANGE
z=0, v, A, yc/q, n/S/yn, Fr/L, M, y_sub → MIXED_INPUT_ENCODING
z>10 or y/b outside [0.01, 20] → VALUE_OUT_OF_RANGE
depth, velocity, or depths as mode → MIXED_INPUT_ENCODING or INVALID_MODE
Example: y=1 Q=2 b=2 z=1 → E≈1.02266 m
Validation cases: 4 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence
y=1 Q=2 b=2 z=1 → E ≈ 1.02266 m
mode=discharge E≈1.02266 y=1 b=2 z=1 → Q ≈ 2 m³/s
y=1 Q=2 b=2 z=0 → MIXED_INPUT_ENCODING
y=1 Q=2 b=2 z=1 v_m_s=1 → MIXED_INPUT_ENCODING
Sources: ISO 80000-4 — Mechanics — Energy, length, speed
Supports: E and y are lengths; Q is volume flow; g is acceleration
Chow, V.T. Open-Channel Hydraulics — Specific energy in a trapezoidal section
Supports: E = y + Q²/(2g A²) with A=(b+z y)y. Rectangular E=y+v²/(2g) is a different page.
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Compute trapezoidal specific energy from depth y, discharge Q, bottom width b, and side slope z.

Supported and not supported

Supported — E from y, Q, b, z · invert Q · API physics.fluid.trapezoid_specific_energy

Not supported — rectangular E=y+v²/(2g); trapezoid yc; trapezoid yn; Manning A,R; Q=Av; trapezoid y invert (that is /calc/physics/trapezoidal-alternate-depth); z=0

Agent / API notes

Capability id: physics.fluid.trapezoid_specific_energy · tool id: trapezoidal-specific-energy · pin 1.0.0.

{ "y": 1, "Q": 2, "b": 2, "z": 1 }

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

Calculator URL stays /calc/physics/trapezoidal-specific-energy. 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 specific energy?

No. E = y + v²/(2g) from y and v is /calc/physics/specific-energy. This page needs z>0 and uses Q, b, z. Sending v to this page, or z to the rectangular page, is MIXED_INPUT_ENCODING. z=0 is MIXED_INPUT_ENCODING.

Is this alternate depth?

No. Both trapezoid y from E, Q, b, and z are /calc/physics/trapezoidal-alternate-depth. Rectangular both y from E, Q, b are /calc/physics/alternate-depth. This page does not invert y. Sending mode=depths or y_sub 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 or q is MIXED_INPUT_ENCODING.

Is this trapezoidal normal depth?

No. Trapezoid yn from Manning is /calc/physics/trapezoidal-normal-depth. Sending n, S, or yn is MIXED_INPUT_ENCODING.

Is this Q = A·v?

No. Volumetric flow from area and velocity is /calc/physics/flow-rate. Sending A or 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 trapezoidSpecificEnergy wrapper.