The Water Level in a Hose: An Accuracy Bound by Length, Sunlight, and Reading Error

A water level — two vertical tubes joined by a water-filled hose — transfers a level datum across distances where a straight sight line is blocked, using the fact that a still water surface in connected vessels settles at one height. This paper computes an accuracy bound for the instrument from stated assumptions: a reading and meniscus standard deviation of 1.0 mm at each end, a thermal error from one sun-warmed standpipe column (expansion coefficient 2.07 × 10⁻⁴ per kelvin, a 10 K difference over a 1.0 m column giving 2.07 mm), a residual capillary allowance of 0.5 mm, and a trapped-air allowance that grows with hose length when the hose is not purged. The hydrostatic error itself does not grow with hose length; length enters only through trapped air and through settling time. Over a 50 m baseline the typical total bound is 2.004 mm with both ends shaded and the hose purged, 3.056 mm with one end in full sun and the hose purged, and 5.556 mm with one end in sun and the hose unpurged — so the widely quoted ±5 mm over 50 m performance holds under the first two condition sets and fails under the third. The smallest true fall the level can certify over 50 m at approximately 95 percent confidence is 4.007 mm shaded and purged, against 11.113 mm in sun unpurged. Settling time, taken as three time constants of 0.06 minutes per metre, reaches 9.0 minutes at 50 m and caps the useful length: the recommended maximum is 75 m, where the wait is 13.5 minutes, beyond a 15-minute working limit at 83.33 m the wait rather than the physics ends the instrument's usefulness. All results are computed, not measured. SIMULATED, NOT MEASURED.