Civil Engineering and Architecture Practice Exam 2026 – Complete Prep Guide

Manning's equation shows how open-channel discharge depends on cross-sectional area, the hydraulic radius, slope, and roughness. The velocity is proportional to R^{2/3} and S^{1/2}, and inversely proportional to the roughness n: v = (1/n) R^{2/3} S^{1/2}. Since discharge Q is velocity times area, you get Q = v A = (1/n) A R^{2/3} S^{1/2}. This form reflects that larger area or a larger hydraulic radius (a bigger flow area relative to wetted perimeter) increases discharge, while rougher channels (larger n) reduce it; and steeper slopes (larger S) increase discharge, but only with the square-root dependence.

R is the hydraulic radius, defined as A divided by the wetted perimeter P, and S is the slope of the energy grade line (roughly the bed slope). The correct expression matches the standard Manning relation for open-channel flow. The other forms misplace the n factor or swap the exponents, which would not align with the observed behavior of flow in channels.