Hydraulic jump energy loss: If D1 and D2 are the upstream and downstream conjugate depths of a hydraulic jump, the loss of specific head at the jump equals

Introduction / Context:A hydraulic jump in a rectangular channel converts supercritical flow to subcritical flow, dissipating energy. Designers must quantify the head loss across the jump to size stilling basins and appurtenances.

Given Data / Assumptions:

• D1 = supercritical depth upstream of the jump.
• D2 = subcritical (sequent) depth downstream.
• Rectangular prismatic channel, steady flow.

Concept / Approach:From the specific energy equation and momentum (sequent depth relation), the loss of specific head h_L across the jump can be expressed purely in terms of D1 and D2.

Step-by-Step Solution:Specific energy upstream: E1 = D1 + V1^2/(2g).Specific energy downstream: E2 = D2 + V2^2/(2g).Using continuity and momentum to eliminate velocities yields the standard head loss:h_L = E1 - E2 = (D2 - D1)^3 / (4 * D1 * D2).

Verification / Alternative check:Check limiting behavior: If D2 ≈ D1, numerator tends to zero faster than denominator, so h_L → 0, as expected for a vanishing jump.

Why Other Options Are Wrong:

• Squares in numerator (option b) underestimates loss.
• (D2 + D1)^3 (option c) has wrong dependence and gives nonzero loss even when D2 = D1.
• Option d lacks the cubic dependence characteristic of jump loss.

Common Pitfalls:Using depth difference squared instead of cubed; forgetting the product 4 * D1 * D2 in the denominator.

Final Answer:(D2 - D1)^3 / (4 * D1 * D2)