Which expression correctly represents Manning's equation for open-channel flow?

• A. Q = (1/n) A R^(2/3) S^(1/2)
• B. Q = n A R^(2/3) S^(1/2)
• C. Q = (1/n) A R^(1/2) S^(2/3)
• D. Q = (1/n) A R^(2/3) S^(1)

Answer: (A)

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.