---
title: "S-curve: 2-h UH to 3-h UH"
subject: "Engineering Hydrology"
chapter: 5
key: "semester_5/engineering_hydrology:scurve-2h-to-3h-a"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-2h-to-3h-a
site: BCT Engineering
---

# S-curve: 2-h UH to 3-h UH

Engineering Hydrology · Chapter 5: Hydrograph Analysis

- Asked in: 2078 Kartik, 6 marks
- Video: [Numerical on 'S' Curve Hydrograph (SSCJE/GATE/ESE/PSU's/DFCCIL/Railways)-Civil Engineering](https://www.youtube.com/watch?v=ChzzFVxNlm0) — Engineering Ace- By Varun Sir
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-2h-to-3h-a
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-5

## Question

The ordinates of a 2-h UH are given below. Derive the ordinates of a 3-h UH by S-curve method.

| Time (hr) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| Ordinates of 2-h UH | 0 | 12 | 48 | 78 | 114 | 72 | 54 |

## Short answer

S-curve built by lagged addition of 2-h UH; lagged S-curve by 3 h, then difference scaled by 2/3. 3-h UH ordinates at 0,2,4,6,8,10,12 h: 0, 8, 36, 68, 102, 86, 60 m³/s.

## Answer

**S-curve method:** the S-curve (S-hydrograph) is the response to a continuous effective rainfall of intensity 1 cm per 2 h. It is built by adding the 2-h UH ordinates lagged by 2 h each. The S-curve is then lagged by 3 h (the new duration) and subtracted; the difference is the response to 3 h of rain at 1 cm per 2 h, which is scaled by $2/3$ to get 1 cm per 3 h.

$$
U_3(t) = \frac{2}{3}\left[S(t) - S(t-3)\right]
$$

#### Step 1: S-curve
$S(t) = U_2(t) + S(t-2)$

| t (h) | $U_2$ | $S(t)$ |
|---|---|---|
| 0 | 0 | 0 |
| 2 | 12 | 12 |
| 4 | 48 | 12 + 48 = 60 |
| 6 | 78 | 60 + 78 = 138 |
| 8 | 114 | 138 + 114 = 252 |
| 10 | 72 | 252 + 72 = 324 |
| 12 | 54 | 324 + 54 = 378 |

#### Step 2: Lag the S-curve by 3 h
$S(t-3)$ is found by linear interpolation between the S-curve ordinates (for example, $S(1)=6$, $S(3)=36$, $S(5)=99$, $S(7)=195$, $S(9)=288$).

| t (h) | $S(t)$ | $S(t-3)$ | Difference | $U_3=\frac{2}{3}\times$ diff (m$^3$/s) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 2 | 12 | 0 | 12 | 8.0 |
| 4 | 60 | 6 | 54 | 36.0 |
| 6 | 138 | 36 | 102 | 68.0 |
| 8 | 252 | 99 | 153 | 102.0 |
| 10 | 324 | 195 | 129 | 86.0 |
| 12 | 378 | 288 | 90 | 60.0 |

**Answer: 3-h UH ordinates at t = 0, 2, 4, 6, 8, 10, 12 h are 0, 8, 36, 68, 102, 86, 60 m$^3$/s.**

(The given 2-h UH is cut off at 12 h with a non-zero ordinate, so the 3-h UH is computed only for this range.)

## Check yourself

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

1. The S-curve of a 2-h UH is built by:
   A. Adding the 2-h UH ordinates repeatedly lagged by 2 h
   B. Multiplying the UH by 2
   C. Subtracting lagged UHs
   D. Integrating base flow
   Answer: A. S(t) = U2(t) + S(t-2) gives the response to continuous rain. (Key point: S-curve construction)

2. The 3-h UH from the 2-h S-curve is:
   A. U3 = (3/2)[S(t) - S(t-3)]
   B. U3 = (2/3)[S(t) - S(t-3)]
   C. U3 = (2/3)[S(t) + S(t-3)]
   D. U3 = S(t) - S(t-2)
   Answer: B. The difference gives 3 h of rain at 1 cm per 2 h; scaling by 2/3 converts to 1 cm per 3 h. (Key point: 3-h UH formula)

3. At t = 4 h, S = 60 and S(t-3) = S(1) = 6. U3(4) is:
   A. 36.0
   B. 54.0
   C. 81.0
   D. 40.0
   Answer: A. (2/3)(60 - 6) = 36 m^3/s. (Key point: Computation)

## Related in Engineering Hydrology

- [S-curve: 2-hr hydrograph from 4-hr UH](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Ascurve-4hr-to-2hr-12-5) — Engineering Hydrology, chapter 5
- [Derive 4-hr UH from observed streamflow](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Auh-4hr-600km2-derive) — Engineering Hydrology, chapter 5
- [Storm hydrograph from 6-h UH, 3-hr storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Astorm-hydrograph-6h-uh-3hr-storm) — Engineering Hydrology, chapter 5, asked 2×
- [S-curve: 4-hr UH to 3-hr UH](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Ascurve-4hr-to-3hr-300km2) — Engineering Hydrology, chapter 5
- [Flood hydrograph from 6-hr UH, two storms](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aflood-hydrograph-3-3-5-5cm) — Engineering Hydrology, chapter 5
- [S-curve 3-h UH and flood from two storms](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Ascurve-2h-3h-flood-8cm-7cm) — Engineering Hydrology, chapter 5

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