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

# Surface profiles on mild slope

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

- Asked in: 2073 Bhadra, 1+4 marks
- Video: [Characteristics of Surface Profile - Mild slope](https://www.youtube.com/watch?v=ctiQfWxDvgA) — Letz Learn
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/mild-slope-profiles-analytical
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-8

## Question

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

## Short answer

A mild slope has yₙ > yc (subcritical uniform flow). M1 curve (y > yₙ) rises downstream, asymptotic to NDL and horizontal at reservoir. M2 (yₙ > y > yc) falls downstream from NDL to CDL steeply. M3 (y < yc) rises downstream, meeting CDL steeply, often causing a hydraulic jump.

## Answer

#### Mild slope

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

#### Analysis

Use the dynamic equation in the wide-channel form

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

with $y_n > y_c$. The NDL and CDL divide the flow into three zones.

**Zone 1: $y > y_n > y_c$ (M1 curve)**

Numerator $> 0$, denominator $> 0$, so $dy/dx > 0$ (depth increases downstream).
- Upstream end: $y \to y_n$, so $dy/dx \to 0$. The curve is **asymptotic to the NDL**.
- Downstream end: $y \to \infty$, so $dy/dx \to S_0$. The water surface becomes **horizontal**, as in a reservoir.
- Backwater curve, caused by dams or weirs.

**Zone 2: $y_n > y > y_c$ (M2 curve)**

Numerator $< 0$, denominator $> 0$, so $dy/dx < 0$ (depth falls downstream).
- Upstream end: $y \to y_n$, $dy/dx \to 0$: **asymptotic to the NDL**.
- Downstream end: $y \to y_c$, denominator $\to 0$, so $dy/dx \to -\infty$: the curve **meets the CDL at a steep angle**.
- Drawdown curve, as at a free overfall or a break to a steeper slope.

**Zone 3: $y < y_c < y_n$ (M3 curve)**

Numerator $< 0$, denominator $< 0$, so $dy/dx > 0$ (depth rises downstream).
- Upstream end: starts from a control at small depth (e.g. below a gate).
- Downstream end: $y \to y_c$, $dy/dx \to \infty$: the curve meets the CDL steeply, and in practice a **hydraulic jump** forms before it.

| Profile | Zone | $dy/dx$ | Upstream end | Downstream end |
|---|---|---|---|---|
| M1 | $y > y_n$ | $+$ | tangent to NDL | horizontal ($S_0$) |
| M2 | $y_n > y > y_c$ | $-$ | tangent to NDL | steeply meets CDL |
| M3 | $y < y_c$ | $+$ | from a control | steeply meets CDL (jump) |

## Check yourself

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

1. A slope is mild when:
   A. yn < yc, S0 > Sc
   B. yn > yc, S0 < Sc
   C. yn = yc
   D. S0 < 0
   Answer: B. On a mild slope uniform flow is subcritical, so yn > yc. (Key point: Definition of mild slope)

2. For the M1 profile (y > yn > yc), dy/dx is:
   A. Negative, falling downstream
   B. Zero
   C. Positive, tangent to NDL upstream and horizontal downstream
   D. Infinite upstream
   Answer: C. Numerator and denominator are both positive; dy/dx tends to 0 at y = yn and to S0 as y tends to infinity. (Key point: M1 curve)

3. For the M2 profile (yn > y > yc), the sign of dy/dx and the downstream end are:
   A. Positive; asymptotic to the NDL downstream
   B. Negative; meets the CDL steeply at the downstream end
   C. Negative; horizontal downstream
   D. Positive; meets the CDL steeply
   Answer: B. Numerator is negative and denominator positive, so depth falls; as y tends to yc, dy/dx tends to minus infinity. (Key point: M2 curve)

4. Why is dy/dx positive on M3 (y < yc)?
   A. Numerator positive and denominator positive
   B. Numerator zero
   C. Both numerator and denominator are negative
   D. S0 is negative
   Answer: C. y < yn makes the numerator negative and y < yc makes the denominator negative, so dy/dx > 0, ending in a jump near yc. (Key point: M3 curve)

## Close variants set in other exams

- [Surface profiles on steep slope](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asteep-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×