Sounding in Fog: The Frequency Band That Carries Across Water, and Why a High-Pitched Boat Whistle Does Not Reach a 600-Metre Crossing

A fog signal succeeds or fails at one distance: the width of the water it must cross. This paper computes that distance from stated assumptions for a river ferry crossing 600 m wide. Received level falls by spherical divergence plus atmospheric absorption, with stated absorption coefficients in clear air from 0.3 dB per km at 125 Hz to 18.0 dB per km at 4000 Hz, and fog adding 0.5 dB per km at 125 Hz but 60.0 dB per km at 4000 Hz. Against an ambient of 55 dB over water in wind and a reliability criterion of 65 dB at the listener, a reference horn of 124 dB at 1 m carries 826.0 m in fog at 125 Hz and 739.1 m at 250 Hz, but only 452.7 m at 1000 Hz and 178.8 m at 4000 Hz; in clear air the same horn reaches 865.0 m at 125 Hz and 394.0 m at 4000 Hz. The band that carries in fog is therefore the low band, roughly 125 to 500 Hz. The bare minimum source for the stated crossing is 121.0 dB at 1 m at 125 Hz, a horn mouth of 0.426 m under the stated sizing relation; with a 6 dB margin the specification becomes a horn of 0.90 m mouth diameter producing 127.5 dB at 1 m, which carries 1199.9 m in fog at 125 Hz. A high-pitched whistle of the kind commonly sold for small boats — 4000 Hz, 115 dB at 1 m — reaches 113.8 m in fog and 206.2 m even in clear air, delivers 12.6 dB at the far bank in fog against the 65 dB criterion, and does not serve as a fog signal for this crossing. SIMULATED, NOT MEASURED.