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

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.

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

Time (hr)024681012
Ordinates of 2-h UH01248781147254

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 2/32/3 to get 1 cm per 3 h.

U3(t)=23[S(t)−S(t−3)]U_3(t) = \frac{2}{3}\left[S(t) - S(t-3)\right]

Step 1: S-curve

S(t)=U2(t)+S(t−2)S(t) = U_2(t) + S(t-2)

t (h)U2U_2S(t)S(t)
000
21212
44812 + 48 = 60
67860 + 78 = 138
8114138 + 114 = 252
1072252 + 72 = 324
1254324 + 54 = 378

Step 2: Lag the S-curve by 3 h

S(t−3)S(t-3) is found by linear interpolation between the S-curve ordinates (for example, S(1)=6S(1)=6, S(3)=36S(3)=36, S(5)=99S(5)=99, S(7)=195S(7)=195, S(9)=288S(9)=288).

t (h)S(t)S(t)S(t−3)S(t-3)DifferenceU3=23×U_3=\frac{2}{3}\times diff (m3^3/s)
00000
2120128.0
46065436.0
61383610268.0
825299153102.0
1032419512986.0
123782889060.0

Answer: 3-h UH ordinates at t = 0, 2, 4, 6, 8, 10, 12 h are 0, 8, 36, 68, 102, 86, 60 m3^3/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.)

✓ Answer checked by Claude Sonnet 5.5

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