Hydraulic jump in a rectangular channel: If q is the discharge per unit width and D1 and D2 are the pre-jump and post-jump water depths respectively, which relationship between q, D1, and D2 holds true for momentum balance across the jump?

Introduction / Context:Hydraulic jumps are rapid transitions from supercritical to subcritical flow that commonly occur in spillways, canal transitions, and energy dissipators. In open-channel hydraulics (civil engineering), their analysis relies on conservation of momentum in a control volume spanning the jump. This question tests recognition of the standard momentum relationship linking discharge per unit width q with upstream and downstream depths D1 and D2 for a rectangular channel.

Given Data / Assumptions:

• Channel is prismatic and rectangular; width b is taken as 1 m so q = Q / b.
• Flow is steady and incompressible, with hydrostatic pressure distribution outside the roller.
• Energy losses in the roller are allowed; momentum (force) balance holds across the jump.

Concept / Approach:The momentum function for a rectangular channel per unit width is M = q^2 / (g * y) + y^2 / 2. Equating M upstream (y = D1) and downstream (y = D2) yields the standard sequent-depth relation in momentum form, from which the required identity for q, D1, and D2 is obtained.

Step-by-Step Solution:

Verification / Alternative check:Dimensional check: q has units m^2/s; q^2/g has m^3. The right-hand side D1D2(D1+D2)/2 also has m^3. The relation is therefore dimensionally consistent.

Why Other Options Are Wrong:

• Option B: Inverts the momentum relation; does not match momentum dimensions.
• Option C: Uses depth difference squared; momentum balance does not produce (D2 - D1)^2 alone.
• Option D: Sums squared depths; missing the multiplicative D1*D2 term from momentum derivation.
• Option E: Harmonic-like form; again misses the (D1 + D2) multiplier in the numerator.

Common Pitfalls:

• Confusing energy conservation with momentum conservation; energy is not conserved across a jump due to losses.
• Forgetting that the momentum function for rectangular channels includes both velocity and hydrostatic terms.
• Using discharge Q instead of unit discharge q without dividing by width.

Final Answer:q^2 / g = (D1 * D2 * (D1 + D2)) / 2