---
title: "S-curve: 2-hr hydrograph from 4-hr UH"
subject: "Engineering Hydrology"
chapter: 5
key: "semester_5/engineering_hydrology:scurve-4hr-to-2hr-12-5"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-4hr-to-2hr-12-5
site: BCT Engineering
---

# S-curve: 2-hr hydrograph from 4-hr UH

Engineering Hydrology · Chapter 5: Hydrograph Analysis

- Asked in: 2080 Baisakh, 8 marks
- Video: [Unit hydrograph of different duration using S-curve (4h to 2h) | Hydrology](https://www.youtube.com/watch?v=5d-M4ov836M) — The Study Sphere
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-4hr-to-2hr-12-5
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-5

## Question

Derive 2-hrs hydrograph from the following 4 hrs unit hydrograph using s-curve method:

| Time (hr) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ordinates of 4 hr UH | 0 | 12.5 | 62.5 | 130 | 175 | 180 | 140 | 90 | 50 | 25 | 13 | 3 | 0 |

## Short answer

Construct the S-curve by adding the4-h UH ordinates lagged every4 h. The2-h UH is then2[S(t)-S(t-2)]. The resulting2-h UH ordinates (m³/s) at0,2,4,…22 h are: 0,25,100,160,190,170,110,70,30,20,6,0. Peak is 190 m³/s at 8 h.

## Answer

**S-curve method.** The S-curve is the hydrograph of a continuous rain of constant intensity. It is built by adding the given 4-h UH repeated every 4 h. A D′-hour UH is then obtained as

$$
U_{D'}(t) = \frac{D}{D'}\,[S(t) - S(t-D')]
$$

with D = 4 h and D′ = 2 h, so the factor is 4/2 = 2.

#### Step 1: Construct the S-curve
Ordinates at 2-h intervals: S(t) = U₄(t) + S(t − 4 h), that is, the 4-h UH added to the S-curve lagged by 4 h (2 steps).

#### Step 2: Lag the S-curve by 2 h and derive the 2-h UH
| t (h) | 4-h UH | S(t) | S(t−2) | 2-h UH = 2[S(t)−S(t−2)] |
|---|---|---|---|---|
| 0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 2 | 12.5 | 12.5 | 0.0 | 25.0 |
| 4 | 62.5 | 62.5 | 12.5 | 100.0 |
| 6 | 130.0 | 142.5 | 62.5 | 160.0 |
| 8 | 175.0 | 237.5 | 142.5 | 190.0 |
| 10 | 180.0 | 322.5 | 237.5 | 170.0 |
| 12 | 140.0 | 377.5 | 322.5 | 110.0 |
| 14 | 90.0 | 412.5 | 377.5 | 70.0 |
| 16 | 50.0 | 427.5 | 412.5 | 30.0 |
| 18 | 25.0 | 437.5 | 427.5 | 20.0 |
| 20 | 13.0 | 440.5 | 437.5 | 6.0 |
| 22 | 3.0 | 440.5 | 440.5 | 0.0 |

The S-curve reaches a constant equilibrium value of 440.5 m³/s (the sum of alternate ordinates of the 4-h UH), so the construction is consistent.

#### Check
Σ(4-h UH) = 12.5 + 62.5 + 130 + 175 + 180 + 140 + 90 + 50 + 25 + 13 + 3 = 881 m³/s, and Σ(2-h UH) = 25 + 100 + 160 + 190 + 170 + 110 + 70 + 30 + 20 + 6 = 881 m³/s. The sums are equal, so both UHs carry the same runoff volume of 1 cm ✓.

**Answer: 2-h UH ordinates (t = 0, 2, 4, … 22 h): 0, 25, 100, 160, 190, 170, 110, 70, 30, 20, 6, 0 m³/s.** Peak = 190 m³/s at t = 8 h.

## Check yourself

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

1. The S-curve is the hydrograph produced by:
   A. A continuous rainfall excess of constant intensity, built by adding the UH repeated at intervals of its duration
   B. A single instantaneous burst of 1 cm
   C. Rainfall that increases uniformly with time
   D. The sum of ordinates of the UH without any lag
   Answer: A. Adding the 4-h UH repeated every 4 h gives the equilibrium (S) hydrograph of continuous rain. (Key point: S-curve meaning)

2. With D = 4 h and D' = 2 h, the 2-h UH ordinate is:
   A. 0.5[S(t) - S(t-2)]
   B. 2[S(t) - S(t-2)]
   C. 2[S(t) - S(t-4)]
   D. [S(t) - S(t-2)]
   Answer: B. U_D'(t) = (D/D')[S(t) - S(t-D')] and D/D' = 4/2 = 2, with S lagged by D' = 2 h. (Key point: Conversion formula)

3. The peak of the derived 2-h UH is:
   A. 180 m3/s at t = 10 h
   B. 175 m3/s at t = 8 h
   C. 190 m3/s at t = 8 h
   D. 160 m3/s at t = 6 h
   Answer: C. At t = 8 h, S(t) = 237.5 and S(t-2) = 142.5, giving 2 x 95 = 190 m3/s. (Key point: Peak result)

4. The S-curve becomes constant at 440.5 m3/s because:
   A. It equals the peak of the UH times 2
   B. It equals the sum of all ordinates of the UH divided by 2
   C. It equals the mean of the UH ordinates
   D. It equals the sum of alternate ordinates of the 4-h UH
   Answer: D. With 2-h steps and a 4-h UH, the repeated additions give a sum of alternate ordinates at equilibrium. (Key point: Equilibrium check)

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