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

# Surface profiles on critical slope

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

- Asked in: 2072 Asoj, 5 marks
- Video: [Classification of channel bed slopes](https://www.youtube.com/watch?v=IjPrXqCwS5c) — WATERCORE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/critical-slope-profiles-analytical
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-8

## Question

Justify analytically the nature of surface profiles in critical sloped channels.

## Short answer

On a critical slope (y_n = y_c), only C1 (y > y_c) and C3 (y < y_c) exist. C1 meets the CDL upstream at a gentle finite slope (dy/dx ≈ 1.11 S0) and becomes horizontal downstream. C3 starts at a control and meets the CDL downstream at the same gentle slope.

## Answer

On a **critical slope**, $S_0 = S_c$ and the normal depth equals the critical depth, $y_n = y_c$. The NDL and CDL coincide, so only two zones exist and zone 2 is absent.

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

**Zone 1: $y > y_n = y_c$ (C1 curve)**

Numerator $> 0$ and denominator $> 0$, so $dy/dx > 0$ (depth rises downstream).
- Downstream end: $y \to \infty$, so $dy/dx \to S_0$: the surface becomes **horizontal**.
- Upstream end: $y \to y_c$. The numerator and denominator both tend to zero, and the limit (with $x = y_c/y \to 1$) is

$$
\lim_{x\to 1}\frac{1 - x^{10/3}}{1 - x^3} = \frac{10/3}{3} = \frac{10}{9}
$$

so $dy/dx \to 1.11\,S_0$ (with the Chezy-type exponent 3 it would be exactly $S_0$). The profile approaches the critical line at a **finite, nearly horizontal angle** (not vertically), and is unstable with surface undulations because $Fr \approx 1$.

**Zone 3: $y < y_n = y_c$ (C3 curve)**

Numerator $< 0$ and denominator $< 0$, so $dy/dx > 0$.
- Upstream end: starts from a control at small depth (e.g. below a gate).
- Downstream end: $y \to y_c$ with the same finite slope $\approx 1.11\,S_0$: it approaches the critical line at a gentle angle, without a jump.

**Zone 2** would lie between $y_n$ and $y_c$, which are equal, so **no C2 curve exists**. Uniform flow at $y_n = y_c$ itself is unstable and is avoided in design.

| Profile | Zone | $dy/dx$ | Upstream end | Downstream end |
|---|---|---|---|---|
| C1 | $y > y_c$ | $+$ | meets $y_c$ at gentle angle | horizontal ($S_0$) |
| C3 | $y < y_c$ | $+$ | from a control | meets $y_c$ at gentle angle |

## Check yourself

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

1. On a critical slope:
   A. yn > yc
   B. yn < yc
   C. yn = yc, so NDL and CDL coincide and no zone 2 exists
   D. yn is infinite
   Answer: C. S0 = Sc gives yn = yc, so there is no space between the lines for a C2 curve. (Key point: Critical slope)

2. Which profiles occur on a critical slope?
   A. C1 and C3 only
   B. C1, C2 and C3
   C. C2 only
   D. None
   Answer: A. Only y > yc (C1) and y < yc (C3) zones exist. (Key point: Profiles that exist)

3. As y tends to infinity on C1, dy/dx tends to:
   A. Zero
   B. Infinity
   C. S0, so the surface becomes horizontal
   D. Minus S0
   Answer: C. Both (yc/y) terms vanish, leaving dy/dx = S0. (Key point: Downstream end of C1)

4. Near y = yc, the indeterminate ratio gives a finite slope (about 1.11 S0), so the curve:
   A. Meets yc vertically
   B. Never approaches yc
   C. Meets yc at a gentle, finite angle
   D. Forms a hydraulic jump always
   Answer: C. The limit (10/3)/3 = 10/9 is finite, so the profile approaches yc at a gentle angle, not vertically. (Key point: Approach to the critical line)

## Related in Hydraulics

- [Surface profiles on mild slope](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Amild-slope-profiles-analytical) — Hydraulics, chapter 8, asked 2×
- [Surface profiles on steep slope](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asteep-slope-profiles-analytical) — Hydraulics, chapter 8, asked 2×
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- [Bed slope for critical normal depth](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Abed-slope-for-critical-flow-at-normal-depth) — Hydraulics, chapter 7
- [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