---
title: "Contraction and bed rise, 3.5 m channel (1 m depth)"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:contraction-bed-rise-3-5m-wide-1m-depth"
source: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-bed-rise-3-5m-wide-1m-depth
site: BCT Engineering
---

# Contraction and bed rise, 3.5 m channel (1 m depth)

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

- Asked in: 2075 Bhadra, 8 marks
- Video: [Open Channel Flow-20 | Channel Width Reduction | Shubham Sarathe](https://www.youtube.com/watch?v=V123w0PH0I8) — EIQ
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-bed-rise-3-5m-wide-1m-depth
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

A 3.5 m wide rectangular channel section carries 4 m³/s of water at a depth of 1 m. If the width is to be reduced to 2.5 m and bed raised by 10 cm, what would be the depth of flow in the contracted section? What maximum rise in the bed level of the contracted section is possible without affecting the depth of flow upstream of the transition?

## Short answer

Depth in the contracted section is y2 = 0.70 m. The maximum bed rise without affecting upstream depth is Δz_max = E1 − Emin2 = 0.108 m (10.8 cm).

## Answer

#### Given data
Upstream $b_1=3.5$ m, $y_1=1$ m, $Q=4$ m³/s; contracted $b_2=2.5$ m with bed raised $\Delta z=0.10$ m.
$$
q_1=\frac{4}{3.5}=1.1429\ \text{m}^2/\text{s},\quad V_1=1.143\ \text{m/s},\quad E_1=1+\frac{1.143^2}{19.62}=1.0666\ \text{m}
$$
$$
q_2=\frac{4}{2.5}=1.6\ \text{m}^2/\text{s},\quad y_{c2}=\sqrt[3]{\frac{1.6^2}{9.81}}=0.6390\ \text{m},\quad E_{min2}=0.9586\ \text{m}
$$

#### Depth in the contracted section
$$
E_2=E_1-\Delta z=1.0666-0.10=0.9666\ \text{m}>E_{min2}=0.9586\ \text{m}
$$
Flow is possible without choking; the flow remains subcritical:
$$
y_2+\frac{1.6^2}{19.62y_2^2}=y_2+\frac{0.13048}{y_2^2}=0.9666\;\Rightarrow\;y_2=0.701\ \text{m}
$$

#### Maximum rise in bed without affecting the upstream depth
The limit is when the contracted section has critical flow with $E_2=E_{min2}$:
$$
\Delta z_{max}=E_1-E_{min2}=1.0666-0.9586=0.1080\ \text{m}\approx10.8\ \text{cm}
$$
**Answer: depth in the contracted section $=0.70$ m; maximum bed rise without affecting the upstream depth $=0.108$ m (about 10.8 cm).**

## Check yourself

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

1. What decides whether the contracted section is choked?
   A. Compare E2 = E1 - Δz with Emin2 for q2 = Q/b2
   B. Compare b2 with b1
   C. Compare y1 with 1 m
   D. Compare V1 with V2
   Answer: A. E2 = 0.9666 m exceeds Emin2 = 0.9586 m, so no choking. (Key point: Check for choking)

2. The depth in the contracted section is the
   A. critical depth 0.639 m
   B. subcritical root of y + 0.13048/y^2 = 0.9666, about 0.70 m
   C. same as upstream 1 m
   D. supercritical root
   Answer: B. Not choked and approach is subcritical, so the subcritical root y2 = 0.701 m is used. (Key point: Depth in the contraction)

3. The maximum bed rise without affecting upstream depth is
   A. E1 + Emin2
   B. Emin2 - E1
   C. E1 - Emin2 = 0.108 m
   D. 0.639 m
   Answer: C. At the limit the contraction has critical flow, E2 = Emin2, so Δzmax = 1.0666 - 0.9586 = 0.108 m. (Key point: Maximum bed rise)

## Close variants set in other exams

- [Critical flow conditions and 3 m to 2 m contraction](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-hump-conditions-critical-flow-and-bed-rise) — Hydraulics, chapter 7, asked 2×

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