Given n as the roughness (rugosity) coefficient, r as hydraulic radius (or hydraulic depth in wide channels), and s as bed slope, which resistance formula gives the mean velocity of flow in m/s?

Introduction / Context:
Engineers often use empirical resistance equations to compute mean velocity in open-channel flow. Recognizing which variables belong to which formula is fundamental.

Given Data / Assumptions:

• Variables provided: n (roughness), r (hydraulic radius/depth), s (slope).
• We must identify the formula that directly gives V in terms of n, r, and s.

Concept / Approach:
Manning’s formula provides velocity directly as V = (1/n) * r^(2/3) * s^(1/2). Chezy's formula uses V = C * sqrt(r * s), where C is separate; Bazin and Kutter predict C, not V directly without Chezy. Froude relations address critical conditions, not resistance.

Step-by-Step Solution:
Match variables (n, r, s) → Manning's equation is the only one with all three directly.Therefore, select Manning’s formula.

Verification / Alternative check:
Check dimensions: r^(2/3) * s^(1/2) gives velocity units when multiplied by 1/n (n is dimensionally adjusted empirically).

Why Other Options Are Wrong:

• Chezy: requires C, not n.
• Bazin/Kutter: give C for Chezy, not V directly in the form shown.
• Froude: relates velocity to depth at critical flow, not roughness.

Common Pitfalls:
Mixing up formulas that estimate C (Bazin/Kutter) with those that yield V directly (Manning).

Final Answer:
Manning's formula: V = (1/n) * r^(2/3) * s^(1/2).