Reading a River to the Nearest Centimetre: A Stilling-Tube Staff Gauge and the Averaging Rule That Certifies It

A staff gauge read against a moving river surface does not show the river's level; it shows the level plus whatever wave happens to be passing. This study certifies two ways of recovering the mean level to within plus or minus 1 cm at 95 percent confidence, under stated oscillation assumptions: wind waves of 0.05 m amplitude and 8 s period, and a surge case of 0.10 m amplitude and 20 s period. For direct reading, a single observation carries a standard deviation of 3.54 cm in the wave case and 7.07 cm in the surge case, and the averaging rule that follows is 49 readings spaced at least 4 s apart (a span of 3.2 minutes) for waves, against 193 readings spaced at least 10 s apart (a span of 32.0 minutes) for surge — a rule so burdensome in the surge case that direct reading must be judged impractical there. For the stilling tube, modelled as a first-order lag from the linearised orifice law with discharge coefficient 0.62, the hole diameter trades damping against lag: in a 200 mm tube the surge case admits only a narrow window, a hole between 11.31 mm and 12.06 mm, meeting both a residual oscillation of at most 1 cm and a lag of at most 2 cm at a flood rise of 2 m per hour; a nominal 12 mm hole sits inside it, with a time constant of 31.99 s, a residual amplitude of 0.99 cm, and a lag of 1.777 cm. At a rise rate of 3 m per hour no hole diameter in any tube size examined meets both targets, because the damping requirement needs a time constant of at least 31.67 s while the lag requirement allows at most 24.0 s; that incompatibility is a plain negative result of the stated targets, not a defect of a particular tube. All values are simulated from stated assumptions, not measured. SIMULATED, NOT MEASURED.