Finding the Water Level by Ear: How Deep a Well an Echo Can Certify, by Stopwatch and by Phone

The depth to water in a well or borehole can be found from the echo of a sharp sound made at the wellhead: depth is half the speed of sound times the echo delay. This paper computes how accurate that simple method is, for two ways of timing the echo — by ear with a stopwatch, and from a phone recording — using a speed of sound of 340.4 m per second at a stated shaft temperature of 15 °C, a stated shaft-temperature uncertainty of 3 °C, a stated human timing uncertainty of 0.20 s, and a stated phone onset-picking uncertainty of 3 ms. The stopwatch method carries a timing error of 34.04 m at every depth and cannot certify any ordinary well: an echo from 10 m returns in 0.0588 s, far inside human reaction time, and the method would need a depth of 340.9 m before its relative error even fell to 10 percent. The phone method carries an error of 0.51 m at 10 m, 0.52 m at 20 m, 0.58 m at 50 m, and 0.74 m at 100 m, and under a stated round-trip audibility model it can certify a depth to a stated tolerance of plus or minus 1.0 m down to 59.4 m over calm water, or 42.0 m over a rough surface. The method costs essentially nothing: a phone already owned is priced at 0 dollars of added cost, against 12 dollars for the stopwatch that cannot do the job. SIMULATED, NOT MEASURED.