Question

Gradually Varied Flow (GVF) profiles in open channels given in Column 1 are to be matched with the water surface slopes in Column 2 in the table below.

Column 1 (GVF Profile)	Column 2 (Water Surface Slope)
(P) M1	(I) Positive
(Q) M2	(II) Negative
(R) M3	(III) Zero

Which of the following options is/are NOT correct?

• A. (P) - (I); (Q) - (II); (R) - (I)
• B. (P) - (I); (Q) - (II); (R) - (III)
• C. (P) - (II); (Q) - (I); (R) - (I)
• D. (P) - (I); (Q) - (III); (R) - (II)

Hint:

Use the GVF sign rule \(dy/dx=(S_0-S_f)/(1-Fr^2)\): M1 and M3 give a positive slope, M2 gives a negative slope; check that against each option.

Answer 1

The Correct Option is B, C, D

This question checks whether you know the direction in which each classic GVF profile (M1, M2, M3) bends on a mild-slope channel, without leaning on the differential-equation sign test. Think of each profile as a curve that must smoothly connect the actual water surface to the normal depth line and the critical depth line, always moving away from critical depth and approaching normal depth far from a control.

1. M1 profile: this occurs when the depth is above both the normal depth $y_n$ and the critical depth $y_c$, typically upstream of a dam, weir, or any obstruction that backs water up on a mild channel. The water surface must rise from the normal-depth line up to the higher depth caused by the obstruction, so as we move downstream toward the control, the depth is increasing, i.e., the surface climbs in the flow direction. That is a positive slope, matching (P) - (I).
2. M2 profile: this occurs when depth lies between $y_c$ and $y_n$, such as just upstream of a free overfall or a channel entering a steeper reach. Here the water is drawn down from the normal depth toward the critical depth as we approach the control, so the depth decreases in the flow direction. A falling surface in the flow direction is a negative slope, matching (Q) - (II).
3. M3 profile: this occurs when depth is below the critical depth, for example just downstream of a sluice gate on a mild channel, where the flow leaves the gate supercritical and must rise back up to normal depth further downstream (usually through a hydraulic jump). Because the depth increases as we move downstream toward the jump, the surface again rises in the flow direction, so this is also a positive slope, matching (R) - (I), not (III) or (II).

So the single correct combination is (P) - (I), (Q) - (II), (R) - (I), which is option (A). Checking the remaining three: option (B) keeps P and Q right but claims R is zero, which is wrong since M3 is not parallel to the bed. Option (C) swaps the signs of M1 and M2 completely, which is wrong. Option (D) keeps P right but mismatches both Q (claims zero instead of negative) and R (claims negative instead of positive), which is wrong.

Let's summarize:

• M1 and M3 profiles both have a rising (positive) water surface slope in the direction of flow, even though M1 lies above normal depth and M3 lies below critical depth.
• M2 is the only one of the three with a falling (negative) water surface slope.
• Only option (A) states the correct combination, so (B), (C), and (D) are all NOT correct.

The options that are NOT correct are (B), (C), and (D).