At-a-Station Hydraulic Geometry of Eight Large US Rivers from 4,191 USGS Discharge Measurements: Exact Closure of the Exponents, a Proof of the Power-Law Section Result, and Evidence That Velocity Rises Faster Than Manning's Equation Allows

Leopold and Maddock wrote width, mean depth and velocity at a river cross-section as power functions of discharge, w=aQ^b, d=cQ^f, v=kQ^m. Using 4,191 field measurements from the US Geological Survey at eight large rivers (1929–2026), I fit the three relations and show why least squares on logarithms gives b+f+m=1 and ack=1 exactly when depth and velocity are defined as A/w and Q/A. I then prove that for a channel whose width grows as a power p of depth, with Manning's equation and constant roughness and slope, b=p/(p+5/3), f=1/(p+5/3), m=(2/3)/(p+5/3), and more generally that any velocity law v ∝ d^γ forces m = γf whatever the section shape. The data reject the Manning value γ = 2/3 at seven of the eight rivers: the ratio m/f ranges from 0.597 (Sacramento River at Verona) to 4.70 (Colorado River at Lees Ferry), with a median of 1.385. Only the Missouri at Hermann (0.670) matches. Velocity rising faster than depth allows under constant roughness means that effective resistance falls, or the energy slope rises, as flow increases. Design that extrapolates velocity from depth with a fixed Manning n can therefore misjudge flood velocities by a wide margin.