Tidal Range Phase Difference Calculator

Tidal range is the vertical distance between successive high and low water, and half that value is the amplitude A of the equivalent sinusoid. The midpoint of the two levels approximates mean sea level, the datum about which the oscillation swings. Treating the tide as one harmonic constituent gives h(t) = MSL + A cos(omega t), where omega = 2 pi / T is the angular frequency. For the principal lunar semidiurnal constituent M2 the period is 12.4206 hours, so omega is 0.5059 radians per hour and one full cycle spans 360 degrees; the principal solar constituent S2 has a period of exactly 12.000 hours, while the diurnal K1 and O1 run at 23.93 and 25.82 hours. Phase difference is simply omega multiplied by the time offset between an observation and a reference high water, which is how tide tables express the lag of one port behind another. Practitioners classify coasts by range using the scheme of J. L. Davies: microtidal below 2 m, mesotidal 2 to 4 m, macrotidal above 4 m, with the Bay of Fundy exceeding 16 m. The spring-neap beat repeats every 14.76 days and is approximated here as roughly 1.2 times mean range at springs and 0.7 at neaps. The limits are real. A single sinusoid cannot represent shallow-water overtides such as M4 and M6, which make flood and ebb durations unequal in estuaries, nor storm surge, river discharge or barometric setup. Official predictions use 37 or more constituents fitted to at least a year of gauge record.

Systematic tidal prediction began with harmonic analysis. William Thomson, later Lord Kelvin, proposed in 1867 that an observed tide curve could be decomposed into sinusoids whose frequencies come from astronomy while their amplitudes and phases are fitted locally. Working through the British Association tidal committee from 1867, he had a mechanical tide-predicting machine built in 1872 and 1873 with the engineer Edward Roberts and the instrument maker Alexander Lege; it summed constituents with pulleys and a wire, drawing a year of tide curve in hours. George Howard Darwin extended the constituent development in the 1880s and introduced much of the notation, including the symbols M2, S2, K1 and O1, still used today. Arthur Thomas Doodson at the Liverpool Tidal Institute, founded in 1919, carried the expansion much further, publishing in 1921 a development of the tide-generating potential into several hundred constituents and the six-integer Doodson numbers that index them. The Doodson-Lege machine served British and Allied planning, including tidal predictions for the Normandy landings in 1944. In the United States, Rollin Harris and E. G. Fischer designed Tide Predicting Machine No. 2, which ran 37 constituents for the Coast and Geodetic Survey from 1910 until digital computers replaced it in 1965. Davies proposed the microtidal, mesotidal and macrotidal classification in 1964, and NOAA and the UK Hydrographic Office still publish predictions built on the same harmonic framework.