At-a-Station Hydraulic Geometry: Channel Width from Discharge

A river does not widen in proportion to its flow. At a given gauging station, the width of the water surface grows only slowly as the discharge rises, because the banks are steep. This relation captures that with a power law, $w = a\,Q^{\,b}$, where $w$ is the surface width, $Q$ is the discharge, and $a$ and $b$ are numbers fitted to the station's own measurements. It is one of three laws, with the depth and the velocity, that together carry the flow.

This is an empirical fit, not a derived law. The coefficient $a$ is the width at a discharge of 1 m³/s, and the exponent $b$ says how quickly the width responds as the flow changes. Fitted exponents typically fall between 0.1 and 0.5, with a mean near 0.26 in Leopold and Maddock's original study. A small exponent describes a channel that barely widens as it fills; a larger one describes a channel that spreads more readily. Both numbers come from one station's field data, so they should not be carried over to another station.

Take $a = 2.5$, $b = 0.26$ and $Q = 100$ m³/s. The page gives a width of 8.278 m, compared with the 2.5 m the same station has at 1 m³/s. The relation can be rearranged to fit a station from measurements: with 8.278 m at 100 m³/s and $a = 2.5$, the exponent is $b = \frac{\ln(w / a)}{\ln Q} = 0.26$, and the same numbers give back 100 m³/s or a coefficient of 2.5 when either is the unknown.

The exponents of the width, depth and velocity laws at a station sum to one, which reflects that together they must carry the whole flow. The sources are Knighton (1998), which treats width, depth and velocity as power functions of discharge, and Dingman (2009), on hydraulic geometry. Being a fit, the relation should be used within the range of flows the station was measured over. Once the width and depth are known, the hydraulic radius follows from them, and the Froude number says whether the flow at that width and depth is slow and deep or fast and shallow.