---
title: "Define hydraulic slope"
subject: "Hydraulics"
chapter: 5
key: "semester_4/hydraulics:define-hydraulic-slope"
source: https://www.bctengineer.com/q/semester_4/hydraulics/define-hydraulic-slope
site: BCT Engineering
---

# Define hydraulic slope

Hydraulics · Chapter 5: Basics of Open channel flow

- Asked in: 2072 Asoj, 1 marks
- Video: [Classification of Channel Bed Slope | Gradually Varied Flow | Hydraulics and Fluid Mechanics](https://www.youtube.com/watch?v=YLtEFjQ-jm4) — The Content Paradox
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/define-hydraulic-slope
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-5

## Question

Define hydraulic slope.

## Short answer

Hydraulic slope (Sw) is the slope of the water surface in an open channel, equal to the fall of piezometric head per unit length: Sw = -d(z+y)/dx. In uniform flow it equals bed slope and energy slope; in non-uniform flow it differs from both.

## Answer

**Hydraulic slope** (hydraulic gradient, $S_w$) is the slope of the hydraulic grade line (HGL), which in an open channel coincides with the water surface. It is the fall of the water surface (piezometric head $z+y$) per unit length of the channel:
$$
S_w=-\frac{d(z+y)}{dx}
$$
For uniform flow the water surface is parallel to the bed, so $S_w=S_0$ (bed slope) and also equals the energy slope $S_f$. For non-uniform flow $S_w$ differs from both $S_0$ and $S_f$.

```text
 EGL ------\
 HGL(WSL)    ------\      S_w = slope of water surface
 Bed             ------\  S_0 = slope of bed
```

## Check yourself

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

1. What is hydraulic slope?
   A. The slope of the channel bed
   B. The slope of the energy grade line
   C. The slope of the hydraulic grade line, i.e. the water surface in an open channel
   D. The ratio of depth to width
   Answer: C. Hydraulic slope S_w is the fall of the water surface (z + y) per unit length. (Key point: Definition)

2. Which expression gives the hydraulic slope?
   A. S_w = -d(z+y)/dx
   B. S_w = -dz/dx
   C. S_w = -dH/dx with velocity head
   D. S_w = dy/dx only
   Answer: A. S_w is the negative gradient of the piezometric head z + y. (Key point: Formula)

3. In uniform flow, how do the slopes compare?
   A. S_w is greater than S0
   B. S_w = S0 = Sf
   C. S_w = 0
   D. Sf is zero
   Answer: B. In uniform flow the bed, water surface and energy line are parallel, so all three slopes are equal. (Key point: Uniform flow)

## Related in Hydraulics

- [Bed, hydraulic and energy slopes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Abed-hydraulic-energy-slopes-not-parallel) — Hydraulics, chapter 5
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