--- title: "S-hydrograph from rectangular 1-hr UH" subject: "Engineering Hydrology" chapter: 5 key: "semester_5/engineering_hydrology:scurve-rectangular-1hr-uh" source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-rectangular-1hr-uh site: BCT Engineering --- # S-hydrograph from rectangular 1-hr UH Engineering Hydrology · Chapter 5: Hydrograph Analysis - Asked in: 2070 Chaitra, 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-rectangular-1hr-uh - Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-5 ## Question A 1 hour unit hydrograph is given by a rectangle whose base is 4 hours and height is 0.25/hour. Construct an S-hydrograph using this UH. ## Short answer Adding successive 1-h rectangular UHs (height0.25/h, base4 h) lagged by1 h: S-curve ordinates rise linearly to0.25,0.50,0.75,1.00 at1-4 h, then remain constant at equilibrium1.00 h⁻¹ for t≥4 h. ## Answer The S-hydrograph (S-curve) is the hydrograph due to a continuous rain of 1 cm/h indefinitely. It is obtained by adding 1-h UHs, each lagged by 1 h: $$ S(t) = U(t) + U(t-1) + U(t-2) + \dots $$ The 1-h UH is a rectangle: height 0.25 per hour, base 4 h (area = 0.25 × 4 = 1 cm ✓). Between 0 and 1 h it has ordinate 0.25, between 1 and 2 h 0.25, and so on up to 4 h. #### Construction (ordinates over each hour) | Time interval (h) | Number of UHs active | S-curve ordinate (per h) | |---|---|---| | 0 – 1 | 1 | 0.25 | | 1 – 2 | 2 | 0.50 | | 2 – 3 | 3 | 0.75 | | 3 – 4 | 4 | 1.00 | | 4 – 5 | 4 (the first UH has ended, the fifth starts) | 1.00 | | > 4 | 4 | 1.00 (equilibrium) | Since each UH rectangle lasts only 4 h, the maximum number of overlapping UHs is 4, so the S-curve reaches its equilibrium value 1.00 at t = 4 h (= base time of the UH) and stays there. ```text S (per h) 1.00 | ___________________ | / 0.75 | / 0.50 | / 0.25 | / 0 +--+--+--+--+--+--+--+-- t (h) 0 1 2 3 4 5 6 ``` **Answer: S(t) = 0.25 t for 0 ≤ t ≤ 4 h (0.25, 0.50, 0.75, 1.00 at the ends of hours 1 to 4), then constant at 1.00 per hour.** The equilibrium value equals 1 cm/h, the intensity of the continuous rain, as it should. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the S-curve built from a 1-h UH? A. Add 1-h UHs each lagged by 1 h B. Subtract successive UHs lagged by 1 h C. Multiply the UH by time D. Integrate the UH once and halve it Answer: A. S(t) = U(t) + U(t-1) + U(t-2) + ... (Key point: S-curve construction) 2. The UH is a rectangle of base 4 h. When does the S-curve reach equilibrium? A. At t = 1 h B. At t = 4 h, the UH base time, when 4 UHs overlap C. At t = 2 h D. Never; it keeps rising Answer: B. The maximum overlap is 4 rectangles, reached at t = 4 h. (Key point: Equilibrium time) 3. What is the equilibrium ordinate (0.25 x 4)? A. 0.25 per hour B. 4.00 per hour C. 1.00 per hour, equal to the 1 cm/h continuous rain intensity D. 0.50 per hour Answer: C. 4 x 0.25 = 1.00, matching the unit intensity. 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