---
title: "Surface profiles on steep slope"
subject: "Hydraulics"
chapter: 8
key: "semester_4/hydraulics:steep-slope-profiles-analytical"
source: https://www.bctengineer.com/q/semester_4/hydraulics/steep-slope-profiles-analytical
site: BCT Engineering
---

# Surface profiles on steep slope

Hydraulics · Chapter 8: Non-uniform gradually varied flow (GVF)

- Asked in: 2073 Magh, 1+4 marks
- Video: [E3: Steep Slope profiles and practical examples](https://www.youtube.com/watch?v=EJmWwAI7M9g) — udit bhatia
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/steep-slope-profiles-analytical
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-8

## Question

What is a steep slope? Justify analytically the nature of surface profiles (both upstream and downstream end) for steep slope.

## Short answer

A steep slope has y_n < y_c (S0 > Sc). The S1 curve rises from a hydraulic jump at the CDL and becomes horizontal downstream; S2 falls steeply from a break in grade and asymptotes to the NDL; S3 rises from a control and also asymptotes to the NDL downstream.

## Answer

#### Steep slope

A channel slope is **steep** when the normal depth is smaller than the critical depth, $y_n < y_c$ (equivalently $S_0 > S_c$). Uniform flow on it is supercritical.

#### Analysis

$$
\frac{dy}{dx} = S_0\,\frac{1 - (y_n/y)^{10/3}}{1 - (y_c/y)^3},\qquad y_c > y_n
$$

**Zone 1: $y > y_c > y_n$ (S1 curve)**

Numerator $> 0$, denominator $> 0$, so $dy/dx > 0$.
- Upstream end: $y \to y_c$, $dy/dx \to \infty$. The curve **meets the CDL at a steep angle**, i.e. it begins at a hydraulic jump.
- Downstream end: $y \to \infty$, $dy/dx \to S_0$; the surface becomes **horizontal**.
- It is a backwater curve behind a dam, weir or sluice gate on a steep slope.

**Zone 2: $y_c > y > y_n$ (S2 curve)**

Numerator $> 0$, denominator $< 0$, so $dy/dx < 0$ (depth falls downstream).
- Upstream end: $y \to y_c$, $dy/dx \to -\infty$: steep departure from the CDL (e.g. from a break from mild to steep slope).
- Downstream end: $y \to y_n$, $dy/dx \to 0$: **asymptotic to the NDL**.

**Zone 3: $y < y_n < y_c$ (S3 curve)**

Numerator $< 0$, denominator $< 0$, so $dy/dx > 0$ (depth rises downstream).
- Upstream end: starts at a control at small depth (below a sluice gate).
- Downstream end: $y \to y_n$, $dy/dx \to 0$: **asymptotic to the NDL**.

| Profile | Zone | $dy/dx$ | Upstream end | Downstream end |
|---|---|---|---|---|
| S1 | $y > y_c$ | $+$ | steeply meets CDL (jump) | horizontal ($S_0$) |
| S2 | $y_c > y > y_n$ | $-$ | steep from CDL | tangent to NDL |
| S3 | $y < y_n$ | $+$ | from a control | tangent to NDL |

## Check yourself

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

1. A steep slope has:
   A. yn > yc and subcritical uniform flow
   B. yn = yc
   C. yn < yc and supercritical uniform flow
   D. No normal depth
   Answer: C. S0 > Sc means normal depth is below critical depth. (Key point: Definition of steep slope)

2. For S1 (y > yc > yn), the ends of the curve are:
   A. Tangent to NDL upstream, horizontal downstream
   B. Steep at CDL upstream (begins at a jump), horizontal downstream
   C. Horizontal upstream, steep downstream
   D. Both ends tangent to NDL
   Answer: B. As y tends to yc, dy/dx tends to infinity; as y tends to infinity, dy/dx tends to S0. (Key point: S1 curve)

3. S2 (yc > y > yn) has numerator positive and denominator negative, so:
   A. Depth rises downstream
   B. It is horizontal at the end
   C. It never reaches yn
   D. Depth falls downstream and is asymptotic to the NDL
   Answer: D. dy/dx < 0, so depth drops from yc toward yn, approaching it tangentially. (Key point: S2 curve)

4. S3 (y < yn) rises downstream because:
   A. Numerator and denominator are both negative, and it approaches yn asymptotically
   B. Numerator positive, denominator negative
   C. S0 is negative
   D. It meets yc steeply
   Answer: A. Both factors are negative so dy/dx > 0, and as y tends to yn, dy/dx tends to 0. (Key point: S3 curve)

## Close variants set in other exams

- [Surface profiles on mild slope](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Amild-slope-profiles-analytical) — Hydraulics, chapter 8, asked 2×

## Related in Hydraulics

- [Surface profiles on critical slope](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acritical-slope-profiles-analytical) — Hydraulics, chapter 8
- [Sketch profiles with upstream control and gate](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asketch-profiles-slope-arrangement-uc-ac) — Hydraulics, chapter 8
- [Sluice gate profiles and lengths by direct step](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asluice-gate-profiles-length-direct-step) — Hydraulics, chapter 8
- [Three successive slopes: profiles and length](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Athree-reaches-profiles-and-length-slope-1) — Hydraulics, chapter 8
- [Locate control points and sketch profiles](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asketch-profiles-three-slope-figures-control-points) — Hydraulics, chapter 8, asked 2×