---
title: "Froude number ratio for alternate depths"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:alternate-depths-froude-number-ratio"
source: https://www.bctengineer.com/q/semester_4/hydraulics/alternate-depths-froude-number-ratio
site: BCT Engineering
---

# Froude number ratio for alternate depths

Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow

- Asked in: 2076 Baisakh, 4 marks
- Video: [Open Channel 1b - Froude number](https://www.youtube.com/watch?v=XNaRJOZyKac) — CE340Hydraulics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/alternate-depths-froude-number-ratio
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

In a rectangular channel, $F_1$ and $F_2$ are the Froude's numbers corresponding to the alternate depths at a certain discharge. Show that: $\left(\frac{F_2}{F_1}\right)^{\frac{2}{3}} = \frac{2+F_2^2}{2+F_1^2}$

## Short answer

At alternate depths, equal specific energy gives y1(2+F1²)=y2(2+F2²). With constant q, (F2/F1)^(2/3) = y1/y2. Substituting yields (F2/F1)^(2/3) = (2+F2²)/(2+F1²).

## Answer

**Alternate depths** $y_1$ and $y_2$ have the same specific energy and the same discharge per unit width $q$.

#### Express $E$ in terms of Froude number
For a rectangular channel, $Fr=\dfrac{V}{\sqrt{gy}}=\dfrac{q}{\sqrt g\,y^{3/2}}$, so
$$
\frac{V^2}{2g}=\frac{Fr^2\,y}{2}
$$
$$
E=y+\frac{Fr^2y}{2}=\frac{y\,(2+Fr^2)}{2}
$$

#### Equal specific energy
$$
\frac{y_1(2+F_1^2)}{2}=\frac{y_2(2+F_2^2)}{2}\;\Rightarrow\;\frac{y_1}{y_2}=\frac{2+F_2^2}{2+F_1^2}\qquad(1)
$$

#### Ratio of Froude numbers
Since $q$ is the same for both depths,
$$
\frac{F_2}{F_1}=\frac{q/(\sqrt g\,y_2^{3/2})}{q/(\sqrt g\,y_1^{3/2})}=\left(\frac{y_1}{y_2}\right)^{3/2}
$$
$$
\left(\frac{F_2}{F_1}\right)^{2/3}=\frac{y_1}{y_2}\qquad(2)
$$

#### Result
From (1) and (2):
$$
\boxed{\left(\frac{F_2}{F_1}\right)^{2/3}=\frac{2+F_2^2}{2+F_1^2}}
$$
Hence proved.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a rectangular channel, E can be written as
   A. y(1 + Fr^2)
   B. y(2 + Fr^2)/2
   C. y(2 - Fr^2)/2
   D. y Fr^2/2
   Answer: B. V^2/2g = Fr^2 y/2, so E = y + Fr^2 y/2 = y(2 + Fr^2)/2. (Key point: E in terms of Froude number)

2. Which quantities are the same for two alternate depths?
   A. Specific energy and specific force
   B. Depth and velocity
   C. Specific energy and discharge per unit width
   D. Froude number and depth
   Answer: C. Alternate depths have equal E and equal q, but different y, V and Fr. (Key point: Alternate depth property)

3. Since Fr = q/(sqrt(g) y^1.5) with the same q, F2/F1 equals
   A. y1/y2
   B. (y1/y2)^(2/3)
   C. (y2/y1)^(3/2)
   D. (y1/y2)^(3/2)
   Answer: D. F2/F1 = (y1/y2)^(3/2), hence (F2/F1)^(2/3) = y1/y2. (Key point: Froude ratio vs depth ratio)

4. Equating E for the two depths gives y1/y2 = (2 + F2^2)/(2 + F1^2). How is the required result obtained?
   A. Substitute y1/y2 = (F2/F1)^(2/3) into this relation
   B. Differentiate with respect to y
   C. Set F1 = F2
   D. Use the jump equation
   Answer: A. Both expressions equal y1/y2, so (F2/F1)^(2/3) = (2 + F2^2)/(2 + F1^2). (Key point: Combining the results)

## Related in Hydraulics

- [Program for alternate depths in rectangular channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aalgorithm-alternate-depths-rectangular) — Hydraulics, chapter 7
- [Specific energy from alternate depths](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aspecific-energy-in-terms-of-alternate-depths) — Hydraulics, chapter 7
- [Hydraulic jump definition and downstream Froude number](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahydraulic-jump-what-is-and-froude-after-jump) — Hydraulics, chapter 9
- [Classification of hydraulic jump by Froude number](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahydraulic-jump-classification-froude) — Hydraulics, chapter 9

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