Learn: Critical depth — rectangular channel
For a rectangular channel carrying a given discharge per unit width, critical depth is the depth at which specific energy is minimum. It separates the two ideal depth branches of open-channel flow.
What the formula is saying
Use q = Q/b, square q, divide by g, then take the cube root. The critical-flow condition v² = gy together with v = q/y leads to y³ = q²/g.
Read the symbols in plain language
- q
- Discharge per unit width
Discharge per unit width. Combining continuity with the critical Froude condition leaves depth cubed.
m²/sUse m²/s as the base unit shown here. q is discharge per unit channel width in m²/s, not total discharge in m³/s. q²/g has unit m³; its cube root gives critical depth in m.
- g
- Gravity
Gravity. Combining continuity with the critical Froude condition leaves depth cubed.
m/s²Use m/s² as the base unit shown here. q is discharge per unit channel width in m²/s, not total discharge in m³/s. q²/g has unit m³; its cube root gives critical depth in m.
- yc
- Result to find
Critical depth — rectangular channel. A cube root converts the cubic length into the required water depth.
m
Sort out the units first
q is discharge per unit channel width in m²/s, not total discharge in m³/s. q²/g has unit m³; its cube root gives critical depth in m.
Assumptions before calculating
Assume a rectangular section, hydrostatic pressure, a uniform velocity approximation and positive discharge magnitude. Width must already have been accounted for in q.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find yc and explain the result in the stated output unit.
- q · Discharge per unit width
- 2 m²/s
- g · Gravity
- 9.81 m/s²
Form the critical-depth cube
Combining continuity with the critical Froude condition leaves depth cubed.
(2)^2 ÷ (9.81) ≈ 0.4077471967 m³Take the cube root
A cube root converts the cubic length into the required water depth.
(0.4077471967)^(1 ÷ 3) ≈ 0.7415327354 m
Does this worked answer make sense?
Substituting yc into v = q/yc should give Fr = v/√(g yc) close to one. Multiplying q by eight makes critical depth four times as large.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- q · Discharge per unit width
- 3 m²/s
- g · Gravity
- 9.81 m/s²
Form the critical-depth cube
Combining continuity with the critical Froude condition leaves depth cubed.
(3)^2 ÷ (9.81) ≈ 0.9174311927 m³Take the cube root
A cube root converts the cubic length into the required water depth.
(0.9174311927)^(1 ÷ 3) ≈ 0.9716827674 m
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- q · Discharge per unit width
- 1.5 m²/s
- g · Gravity
- 9.81 m/s²
Find: Learn: Critical depth — rectangular channel
A hint, not the answer
Use q = Q/b, square q, divide by g, then take the cube root. The critical-flow condition v² = gy together with v = q/y leads to y³ = q²/g.
q is discharge per unit channel width in m²/s, not total discharge in m³/s. q²/g has unit m³; its cube root gives critical depth in m.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Form the critical-depth cube
Combining continuity with the critical Froude condition leaves depth cubed.
(1.5)^2 ÷ (9.81) ≈ 0.2293577982 m³Take the cube root
A cube root converts the cubic length into the required water depth.
(0.2293577982)^(1 ÷ 3) ≈ 0.6121217863 m
Avoid the common trap
Do not use total Q without dividing by width. Take a cube root, not a square root, and do not confuse critical depth with critical hydraulic radius.
When this method applies — and when it does not
This explicit formula is not valid unchanged for trapezoidal or circular open channels. It gives critical depth, not normal depth from resistance or the depth that must actually occur at a given site.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Critical depth — rectangular channel. Water-measurement principles; for discharge devices, read the orifice/weir chapters and the installation and head-measurement conditions, not only the coefficient formula.
- U.S. Bureau of Reclamation — Water Measurement Manual, 3rd edition (1997; revised reprint 2001)
- Dawei Han, University of Bristol — Concise Hydraulics (2008, Ventus Publishing)
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.