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 Engine tested · Source checked · 5/5 tests · Production surface contract 1/1 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 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
Sources
Evidence: 5 property · Production surface contract 1/1 · Artifact integrity PASS
Production: Embedded snapshot: STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · 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 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

Enter E, Q, and b

Specific energy, discharge, and rectangular width. 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=1.05099 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

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

Engine tested 5/5 tests · Production surface contract 1/1
Named expert review Not performed
Calculation version 1.0.0

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 · 5/5 tests · Production surface contract 1/1 · 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

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: Specific energy, Specific force, Critical depth, Hydraulic jump, Froude number .

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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.

Where does this run?

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