---
title: "Why critical depth varies with constriction only"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:critical-depth-varies-constriction-not-hump"
source: https://www.bctengineer.com/q/semester_4/hydraulics/critical-depth-varies-constriction-not-hump
site: BCT Engineering
---

# Why critical depth varies with constriction only

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

- Asked in: 2068 Bhadra, 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/critical-depth-varies-constriction-not-hump
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

Why does the critical depth vary for the constriction flow analysis and not vary for the hump flow analysis?

## Short answer

Critical depth y_c = (q^2/g)^(1/3) depends on discharge per unit width q = Q/b. A hump leaves the width unchanged, so q and y_c remain constant. A constriction reduces width, increasing q and thus y_c in the throat, so the critical depth differs between the approach and the contracted section.

## Answer

For a rectangular channel the critical depth depends only on the **discharge per unit width**:

$$
y_c = \left(\frac{q^2}{g}\right)^{1/3},\qquad q = \frac{Q}{b}
$$

- **Hump (bed raised):** the width $b$ stays the same, so $q$ is unchanged. Hence $y_c$ is the same at every section. A hump only changes the specific energy available above its crest ($E - \Delta z$). When $E - \Delta z$ falls to $E_c = 1.5\,y_c$, the flow becomes critical on the crest with the same $y_c$.
- **Constriction (width reduced):** the width changes from $b_1$ to $b_2$, so $q$ increases to $Q/b_2$. A larger $q$ gives a larger $y_c$ and larger $E_c$ in the throat. The critical depth therefore differs from section to section.

So $y_c$ varies with a constriction because $q$ varies; it does not vary with a hump because $q$ does not.

## Check yourself

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

1. For a rectangular channel, critical depth depends only on:
   A. Bed slope and roughness
   B. Discharge per unit width q (and g)
   C. Upstream velocity
   D. Specific energy at the section
   Answer: B. yc = (q^2/g)^(1/3), so only q matters. (Key point: What yc depends on)

2. Why does yc stay the same over a hump?
   A. The hump lowers the energy to zero
   B. The depth is always critical
   C. Width and hence q do not change
   D. The velocity is constant
   Answer: C. A hump does not change width, so q and yc are unchanged; only the available energy E - dz changes. (Key point: Hump case)

3. Why does yc differ in a constricted throat?
   A. The width falls, so q = Q/b rises and yc and Ec rise
   B. The bed is raised
   C. Q is reduced by the constriction
   D. Friction increases
   Answer: A. Smaller b gives larger q, hence larger yc and Ec at the throat. (Key point: Constriction case)

## Related in Hydraulics

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