The unit hydrograph theory is based on the principle of linearity and time invariance. Which of the following is a direct consequence of this principle?
- AThe duration of a unit hydrograph is always one hour
- BThe peak of a hydrograph from 2 cm of runoff will be twice the peak of the unit hydrograph (for 1 cm)Correct
- CA unit hydrograph can only be used for large river basins
- DThe time base of a unit hydrograph changes with the intensity of the storm
Solution
Concept
Linearity and proportionality in unit hydrograph theory
Formula
Q(t) = R × U(t)
Step-by-step solution
1. Sherman’s unit hydrograph theory assumes linear response (proportionality and superposition) and time invariance. 2. Under linear response, multiplying effective rainfall by factor n multiplies direct runoff ordinates by n. 3. A unit hydrograph corresponds to 1 cm of effective runoff depth. 4. An excess rainfall depth of 2 cm produces ordinates exactly twice the unit hydrograph values, doubling the peak discharge. Answer: B
Explanation
Sherman's unit hydrograph theory assumes linear response (proportionality and superposition) and time invariance. Under linear response, the ordinates of the direct runoff hydrograph are directly proportional to the rainfall excess depth. Therefore, an effective rainfall of 2 cm produces hydrograph ordinates and a peak discharge exactly twice those resulting from a unit hydrograph of 1 cm.
Why the other options are wrong
A) Unit hydrograph duration depends on storm duration D and is not fixed to 1 hour. C) Unit hydrographs apply to small to moderate catchments (<5000 km²), not very large river basins. D) Time base remains constant for a specified duration under time invariance.
Shortcut
Linear system = Direct proportionality (2 cm excess rainfall gives 2x peak flow).
Common mistake
Assuming storm intensity alters the hydrograph time base, violating the principle of time invariance.
Correct answer: B) The peak of a hydrograph from 2 cm of runoff will be twice the peak of the unit hydrograph (for 1 cm)