The Tide-Predicting Machine

A Port Needs a Table

Ships come and go by the depth of water, and the depth changes with the tide twice a day. A port needs to know the times and heights of high and low water for every day of the coming year.

The harmonic method writes a port’s tide as a sum of cosines. The period of each comes from astronomy; its amplitude and phase come from analysing that port’s tide-gauge records. As the US Coast and Geodetic Survey later put it, without a predicting machine the harmonic method would involve too much labour to be of practical service.

The catalogue of the 1876 South Kensington exhibition said of Kelvin’s first machine that it could be turned fast enough to run off a year’s tides for any port in about four hours.

Many Rises and Falls, Added

The Moon makes the sea rise and fall once every 12.42 hours: this term is called M2. The Sun’s term takes exactly 12 hours: S2. Now in step and now opposed, they make the range grow and shrink over about 14.8 days: spring tides and neap tides.

The Moon’s eccentric orbit and the Moon and Sun moving off the equator add more terms with slightly different periods: N2, K2 and L2 near half a day, K1, O1 and P1 near a day. Shallow water makes terms of a quarter day, such as M4 and MS4. Kelvin’s first machine carried these ten.

Their sizes differ from port to port. In NOAA’s figures, Boston’s M2 is 1.37 m and its K1 only 0.14 m: two highs and two lows a day. At Pensacola on the Gulf of Mexico, M2 is 1.7 cm while K1 and O1 are about 12 cm each: on most days the tide rises and falls only once.

One Crank per Term

Each term is a turning gear wheel with a crank pin on its face. The pin runs in a horizontal slot on a rod that can only slide up and down: as the wheel turns evenly, the rod moves up and down as a cosine. This is a crosshead and slide, a Scotch yoke.

How far the pin sits from the axis is the term’s amplitude; the angle the wheel is set to before the run is its phase. A new port means setting these twenty numbers again.

On the first machine the pulleys rode on the cranks themselves and moved in circles, with a small geometric error. Kelvin’s later machines used the crosshead and slide, so the motion is an exact cosine.

Tooth Counts Set the Speeds

Every term must turn exactly as fast as it does in the sky. M2 turns 28.9841042 degrees an hour, just short of S2’s 30, and that small difference is what makes springs and neaps.

On the US Coast and Geodetic Survey’s Machine No. 2 every shaft turns 30 degrees an hour, then drives through two pairs of gears: for M2, 30 × 103/74 × 59/85 = 28.9841017 degrees an hour, 0.02 degree out in a year. The tooth counts come from writing each speed ratio as two fractions and picking the nearest pair.

With the cast-iron front frame cut away, the bevel gears on the main shaft hand the motion to every spindle.

M2: 30 × 103/74 × 59/85 = 28.9841017 °/h

One Wire Does the Sum

The cranks stand in two rows, an upper and a lower; each rod ends in a pulley, and all the pulleys lie in one plane. A fine steel wire, fixed at one end, runs over an upper pulley and under a lower one in turn, every free length of it vertical; the other end carries the pen.

When an upper pulley rises, or a lower one falls, the wire round it takes up twice that length, so the hanging pen rises twice as far. With all ten moving, the pen’s height is twice the sum of the ten terms. The idea of a chain over pulleys for adding came from Wheatstone; Kelvin used a wire instead.

A Pen Draws the Year

Kelvin wrote that the hanging weight is an ink bottle with a glass tubular pen, which marks the tide level in a continuous curve on a long band of paper, moved horizontally by a vertical cylinder geared to the shafts.

At about a foot a day, a year’s curve is 365 feet long. Machine No. 2 was cranked at 20 turns a minute, six turns to a machine day, so tracing a year took about 1 hour 50 minutes; with the reading and copying, a port’s year took 10 to 15 hours.

Taken Apart

The first machine was made in 1872–73 by A. Légé and Co. of London: 1.82 m long, 1 m wide, 0.77 m high and 86 kg. It is in the Science Museum, London. The arithmetic of its gearing was Edward Roberts’.

Later machines grew. Machine No. 2 in the United States is about 11 feet long and 6 feet high, carries 37 terms and weighs about 2,500 pounds. It has a second crank on every shaft, set 90 degrees ahead and summed on its own chain, so it stops at the times of high and low water.

“Position Z”, 1944

On 9 October 1943, Commander Farquharson of the Admiralty’s Tidal Branch sent Arthur Doodson at the Liverpool Tidal Institute 11 pairs of harmonic constants for a place named only “Position Z”, and asked for hourly heights for four months from 1 April 1944. It was Normandy.

The landings had to start an hour after first light at low water, so the engineers could see and clear the obstacles on the beach. Low water came at different times on the five beaches: more than an hour apart between Utah and Sword, about 100 km apart. Only 5, 6 and 7 June 1944 met every condition. Doodson used the Institute’s two machines: one of Kelvin’s type and one designed by Roberts.

What It Changed

Only about thirty machines of this kind were ever made, for Britain, the United States, Germany, France, Japan, India, Brazil and others. Every US tide table from 1912 to 1965 came from Machine No. 2; from 1966 electronic computers took over. A German machine built in 1935–39 carried 62 terms.

The machines retired; the method did not. Tide tables are still made by adding a port’s few dozen harmonic constants at the speeds astronomy gives, only now a computer turns the crank.