Critical depth in a rectangular channel in terms of unit discharge q (discharge per unit width): Select the correct expression for y_c.

Introduction / Context:Critical flow in open channels corresponds to Froude number Fr = 1. For a rectangular channel, the critical depth y_c is directly related to the unit discharge q (discharge per unit width). Correctly recalling this formula is fundamental in flow measurement and control structure design.

Given Data / Assumptions:

• Rectangular channel of unit width for unit discharge definition
• Steady, prismatic, frictionless control section
• Acceleration due to gravity g is constant

Concept / Approach:For a rectangular channel, specific energy E = y + v^2/(2g). At critical flow, dE/dy = 0 which leads to v_c^2 = g * y_c. Since q = v * y for unit width, combine to obtain y_c in terms of q and g.

Step-by-Step Solution:

Verification / Alternative check:Dimensional check: [q] = m^2/s for unit width; q^2/g has dimension m^3; cube root gives meters, consistent for depth.

Why Other Options Are Wrong:

• (g / q^2)^(1/3): Inverts the relation; would have dimension 1/m, not a length.
• (q / g)^(1/2): Wrong power and units.
• q / g: Linear in q, dimension m^2/s^2, not a length.

Common Pitfalls:Forgetting that q is per unit width; applying the same formula to non-rectangular sections without the appropriate geometric function; mixing up specific energy and specific force criteria.

Final Answer:y_c = (q^2 / g)^(1/3)