---
title: "Variation of hydraulic radius with depth"
subject: "Hydraulics"
chapter: 5
key: "semester_4/hydraulics:hydraulic-radius-variation-wide-deep-channel"
source: https://www.bctengineer.com/q/semester_4/hydraulics/hydraulic-radius-variation-wide-deep-channel
site: BCT Engineering
---

# Variation of hydraulic radius with depth

Hydraulics · Chapter 5: Basics of Open channel flow

- Asked in: 2079 Chaitra, 2 marks; 2075 Baisakh, 6 marks; 2069 Poush, 2 marks (asked 3 times)
- Video: [Open Channel Flow - 6 [Flow Area A, Wetted Perimeter P Hydraulic Radius R, and Hydraulic Depth D]](https://www.youtube.com/watch?v=lyecuNbxlAs) — Abhijit
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/hydraulic-radius-variation-wide-deep-channel
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-5

## Question

How does hydraulic radius vary for wide and narrow (deep) rectangular channels? Show that hydraulic radius is equal to depth of flow for a wide rectangular channel and half of the bed width for deep gorges.

## Short answer

For a rectangular channel, R = by/(b+2y). When b >> y, R approx y (wide channel). When y >> b, R approx b/2 (deep gorge).

## Answer

**Hydraulic radius** $R=\dfrac{A}{P}$ is the ratio of flow area to wetted perimeter. For a rectangle of bed width $b$ and depth $y$:
$$
A=by,\qquad P=b+2y,\qquad R=\frac{by}{b+2y}
$$

#### Variation
- For a given $b$, $R$ increases with $y$ from zero (at $y=0$) and approaches the limit $b/2$ for very large $y$.
- For a given $y$, $R$ increases with $b$ and approaches $y$ for very large $b$.

| Channel | Condition | $R$ |
|---|---|---|
| Wide, shallow | $b\gg y$ | $\approx y$ |
| Narrow, deep (gorge) | $y\gg b$ | $\approx b/2$ |

#### Wide rectangular channel
Divide numerator and denominator by $b$:
$$
R=\frac{y}{1+2y/b}
$$
For $b\gg y$, $2y/b\to0$, so
$$
R\approx y
$$
The side walls contribute negligible wetted perimeter, so $P\approx b$ and $R=by/b=y$.

#### Deep narrow channel (gorge)
Divide numerator and denominator by $y$:
$$
R=\frac{b}{b/y+2}
$$
For $y\gg b$, $b/y\to0$, so
$$
R\approx\frac{b}{2}
$$
Here the bed is small compared with the two deep walls, so $P\approx2y$ and $R=by/2y=b/2$.

```text
 wide: R ~ y            gorge: R ~ b/2
 ~~~~~~~~~~~~~~~        |~~|
 |_____________|        |  |  y >> b
```

## Check yourself

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

1. Hydraulic radius is
   A. A/T
   B. P/A
   C. A/P
   D. T/A
   Answer: C. Flow area over wetted perimeter. (Key point: Definition)

2. For b much greater than y, R approaches
   A. b
   B. b/2
   C. y
   D. 2y
   Answer: C. P is about b so R = by/b = y. (Key point: Wide channel)

3. For y much greater than b, R approaches
   A. b
   B. y
   C. 2b
   D. b/2
   Answer: D. P is about 2y so R = by/2y = b/2. (Key point: Deep gorge)

4. For fixed b, as y increases R
   A. Decreases
   B. Increases without limit
   C. Increases towards b/2
   D. Stays constant
   Answer: C. R = b/(b/y + 2) approaches b/2. (Key point: Variation with y)

## Related in Hydraulics

- [Contraction and bed rise, 3.5 m channel (1 m depth)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-bed-rise-3-5m-wide-1m-depth) — Hydraulics, chapter 7, asked 2×
- [Contraction and bed rise, 3.5 m channel (1.2 m depth)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-bed-rise-3-5m-wide-1-2m-depth) — Hydraulics, chapter 7
- [Dynamic equation of GVF, wide rectangular form](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Agvf-dynamic-equation-wide-rectangular) — Hydraulics, chapter 8, asked 2×
- [Trapezoidal to rectangular transition with bed lowering](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Atrapezoid-to-rectangular-transition-bed-lowering) — Hydraulics, chapter 7
- [GVF in 12 m wide channel, 2 steps](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Agvf-30m3s-12m-wide-two-steps) — Hydraulics, chapter 8
- [Program for uniform depth in rectangular channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aalgorithm-uniform-depth-rectangular) — Hydraulics, chapter 6

## Related in other subjects

- [Uniform flow in rectangular channel (Chezy)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Arectangular-channel-chezy-flow) — Hydro Power, chapter 1
- [Balancing depth: bed width 20 m](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Abalancing-depth-bed20-cut15-fill2) — Irrigation and Drainage Engineering, chapter 3, asked 3×
- [Balancing depth: bed width 25 m](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Abalancing-depth-bed25) — Irrigation and Drainage Engineering, chapter 3, asked 3×
- [Stable depth and bed width: Q 5, d50 3 cm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Astable-depth-bed-width-q5-d50) — Irrigation and Drainage Engineering, chapter 4