Your job title is computer. Not a machine β a person. It is October 1943, and you have just climbed Bidston Hill in the blackout, past the dark lighthouse, to the sandstone observatory where the Liverpool Tidal Institute calculates the tides for two-thirds of the world's ports. Across the river, barrage balloons hang over bombed Liverpool like tethered whales. Inside, six of you β all young women, plus one famously precise director β turn the mathematics of the Moon into numbers ships can trust.
Your director is Dr. Arthur Thomas Doodson, 53, the world's leading authority on tide prediction. He is profoundly deaf β he built his own hearing aid because none good enough existed β and he is, this morning, holding a letter that he cannot tell you much about. It is handwritten, three pages, urgent, from Commander William Ian Farquharson, Superintendent of Tides at the Admiralty. It asks for hourly tide heights, four months' worth, starting 1 April 1944, for a stretch of enemy coast the letter refuses to name.
The letter calls it only "Position Z."
What you cannot know: you are about to compute the tides for D-Day.
β Click to unseal Commander Farquharson's letter
Hydrographic Dept., Bath β 9th October 1943
Dear Doodson,
We require, with the utmost urgency, hourly predictions of the height of the tide for the four months commencing 1st April 1944, for the position denoted Z in the attached. I regret that "the place is nameless and the constants inferred⦠there is in fact very little data for it."
Enclosed are eleven pairs of harmonic constants. I am gambling on the inferred shallow-water constants giving something like the right answer.
β W. I. Farquharson, Supt. of Tides
Eleven pairs of numbers β an amplitude H and a phase ΞΊ for each "ingredient" of the tide (display reconstructed; the real quoted lines are from Farquharson's actual letter). The eleven names are tidal constituents: the Moon's wave, the Sun's wave, and their echoes in shallow water. Doodson never asked where Position Z was. He guessed β and kept it to himself.
Why couldn't the Admiralty just look the tides up? Because nobody owns a tide table for a nameless beach. And why couldn't they simply say "it's Normandy"? Because if that word leaked, the greatest invasion in history would fail before it began. So the secret travelled as pure mathematics: eleven pairs of numbers, meaningless to a spy, everything to a machine on a hill.
Here is the beautiful secret Doodson's machines are built on. The tide at any port looks hopelessly complicated β high tides that drift later each day, huge "spring" tides, feeble "neap" tides, seasonal moods. But in 1775 Laplace realised, and Lord Kelvin proved useful a century later, that the whole mess is just simple waves added together. Each wave β each constituent β is a perfect cosine: a height traced out by a point going round and round a circle at one fixed, astronomical speed.
h(t) = Aβcos(Οβt + Οβ) + Aβcos(Οβt + Οβ) + Aβcos(Οβt + Οβ) + β¦
The Moon's main wave (Mβ) repeats every 12 h 25 min. The Sun's (Sβ) every 12 h exactly. They slowly slide in and out of step β in step: spring tides. Out of step: neap tides. The equation IS the sea.
Joseph Fourier had shown (1807β1822) that virtually any repeating curve can be built from such waves. Kelvin's genius was to make the addition physical: let a brass wheel turn at each constituent's speed, let each wheel bob a pulley up and down by its amplitude, and thread one continuous wire over all the pulleys. The wire's free end rises and falls by the sum of all the bobbing motions. Tie a pen to it, roll paper past β and the machine draws next year's sea. An equation you can crank.
This is your working replica of the Institute's machines (two of them, kept in separate rooms so one German bomb couldn't destroy both). Switch the sea's ingredients on and off, change their strength, and watch the pen add cosines in brass. Speed up time to see a whole fortnight β the springβneap heartbeat of the ocean β emerge from two circles sliding in and out of step.
Simplified for clarity: the real machines used gear trains, doubled wire runs and 26β40 constituents; Kelvin's 1872β73 original summed 10 and could crank out a year of tides in about four hours. Watch the tag above the paper: when Moon and Sun waves fall into step you get SPRINGS β the huge tides an invasion needs.
Born in 1890, a Lancashire mill manager's son, Doodson went profoundly deaf as a young man β and answered by designing his own hearing aid. Rejected from ordinary war work by his deafness, he became a calculator of extraordinary power; in 1921 he mapped the tide-generating force of Moon and Sun into hundreds of distinct constituents (nearly 400), each with its own code number β Doodson numbers, still used today. By the 1940s his institute at Bidston predicted tides for most of the world's ports. In wartime his staff was down to himself and six young women computers, who calculated by day and fire-watched for incendiary bombs by night. When the nameless letter came, he simply worked out the answer β and never asked the question.
Spring 1944. Your hourly heights for Position Z are in Admiralty hands, and the planners' problem becomes brutally specific. Rommel has sown the beaches with obstacles β steel, concrete and mined timber every two or three yards below the high-water line β designed to rip out the bottoms of landing craft. So the landings must come at low water, just after first light, when the obstacles stand exposed and demolition teams can see them; then on a rising tide (about a metre an hour) so each wave of boats floats free instead of stranding. The airborne divisions, dropping the night before, need a bright moon. Pick a day. The checklist will tell you if the sea agrees.
Dr. Doodson does not tolerate arithmetic errors. Eight questions before you leave the hill.