---
title: "Specific energy from alternate depths"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:specific-energy-in-terms-of-alternate-depths"
source: https://www.bctengineer.com/q/semester_4/hydraulics/specific-energy-in-terms-of-alternate-depths
site: BCT Engineering
---

# Specific energy from alternate depths

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

- Asked in: 2076 Bhadra, 6 marks
- Video: [Specific energy and alternate depth in a rectangular channel | Open Channel Flow](https://www.youtube.com/watch?v=Ok3dSdIMh_w) — The Study Sphere
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/specific-energy-in-terms-of-alternate-depths
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

If $y_1$ and $y_2$ are alternate depths in rectangular channel show that specific energy $E = \frac{y_1^2 + y_1 y_2 + y_2^2}{(y_1 + y_2)}$.

## Short answer

Equate the specific energy expressions for alternate depths y1 and y2, cancel (y2−y1), and substitute the resulting q²/(2g) into E. This gives E = (y1^2 + y1 y2 + y2^2) / (y1 + y2).

## Answer

#### Set-up
For a rectangular channel with discharge $q$ per unit width, two alternate depths $y_1$ and $y_2$ have the same specific energy $E$:
$$
E=y_1+\frac{q^2}{2gy_1^2}=y_2+\frac{q^2}{2gy_2^2}\qquad(1)
$$

#### Eliminate $q$
From the equality in (1):
$$
y_2-y_1=\frac{q^2}{2g}\left(\frac{1}{y_1^2}-\frac{1}{y_2^2}\right)=\frac{q^2}{2g}\,\frac{(y_2-y_1)(y_2+y_1)}{y_1^2y_2^2}
$$
Cancelling $(y_2-y_1)\neq0$:
$$
\frac{q^2}{2g}=\frac{y_1^2y_2^2}{y_1+y_2}\qquad(2)
$$

#### Specific energy
Substitute (2) into $E=y_1+\dfrac{q^2}{2gy_1^2}$:
$$
E=y_1+\frac{y_1^2y_2^2}{(y_1+y_2)y_1^2}=y_1+\frac{y_2^2}{y_1+y_2}
$$
$$
E=\frac{y_1(y_1+y_2)+y_2^2}{y_1+y_2}
$$
$$
\boxed{E=\frac{y_1^2+y_1y_2+y_2^2}{y_1+y_2}}
$$
Hence proved.

## Check yourself

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

1. Alternate depths y1 and y2 are depths with
   A. the same q and the same specific energy E
   B. the same specific force
   C. the same velocity
   D. the same Froude number
   Answer: A. At a given discharge and E, the E-y curve has one subcritical and one supercritical depth. (Key point: Alternate depths definition)

2. Equating the two expressions for E and cancelling (y2 - y1) gives q^2/(2g) equal to
   A. y1 y2/(y1 + y2)
   B. y1^2 y2^2/(y1 + y2)
   C. (y1 + y2)/(y1^2 y2^2)
   D. y1^2 y2^2/(y1 - y2)
   Answer: B. From y2 - y1 = (q^2/2g)(y2 - y1)(y2 + y1)/(y1^2 y2^2), q^2/2g = y1^2 y2^2/(y1 + y2). (Key point: Eliminating q)

3. Substituting into E = y1 + q^2/(2g y1^2) gives
   A. y1 + y2^2/(y1 + y2)
   B. y1 + y2/(y1 + y2)
   C. y1 + y1^2/(y1 + y2)
   D. y1 + y2^2/(y1 - y2)
   Answer: A. The y1^2 cancels, leaving E = y1 + y2^2/(y1 + y2), which simplifies to (y1^2 + y1 y2 + y2^2)/(y1 + y2). (Key point: Substitution step)

## 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
- [Froude number ratio for alternate depths](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aalternate-depths-froude-number-ratio) — Hydraulics, chapter 7
- [Minimum specific energy for triangular channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aminimum-specific-energy-triangular-5-4-yc) — Hydraulics, chapter 7
- [Specific energy, minimum value and critical flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aspecific-energy-minimum-critical-curve) — Hydraulics, chapter 7, asked 2×
- [Specific force and conjugate depths](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aspecific-force-curve-conjugate-depths) — Hydraulics, chapter 7, asked 2×
- [Width equals 8/9 of minimum specific energy](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acritical-flow-min-wetted-perimeter-width-8-9) — Hydraulics, chapter 7

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