Open-channel hydrometry: A stream is 10 m deep. Using the two-point method, a current meter records velocities of 0.7 m/s at 2 m below the surface (0.2D) and 0.3 m/s at 8 m below the surface (0.8D). Compute the discharge per unit width (m^3/s per m).

Introduction / Context:In hydrometry, the velocity distribution over depth is often estimated with point-velocity measurements. The two-point method (at 0.2D and 0.8D, where D is depth) gives a reliable estimate of the mean velocity in verticals for open-channel flow measurements.

Given Data / Assumptions:

• Total depth D = 10 m.
• Point velocities: V(0.2D) = 0.7 m/s at 2 m depth; V(0.8D) = 0.3 m/s at 8 m depth.
• Steady flow and typical velocity profile; negligible secondary currents for this estimation.

Concept / Approach:The two-point method estimates the depth-averaged velocity V_avg in a vertical as the arithmetic mean of velocities at 0.2D and 0.8D. Discharge per unit width q is then the product of V_avg and depth D.

Step-by-Step Solution:V_avg = (V_0.2D + V_0.8D) / 2 = (0.7 + 0.3) / 2 = 0.5 m/sq = V_avg * D = 0.5 * 10 = 5 m^3/s per metre width

Verification / Alternative check:A single-point method (0.6D) typically yields a similar magnitude. Here, the symmetric values around the mean support the average of 0.5 m/s.

Why Other Options Are Wrong:2, 3, 4, and 6 m^3/s per m do not match the calculated q = 5 based on the standard two-point averaging technique.

Common Pitfalls:Confusing depth positions (from surface vs. bed) or averaging the depths instead of velocities; forgetting to multiply the mean velocity by the full depth to obtain discharge per unit width.

Final Answer:5.