A tide-predicting machine was a special-purpose mechanical analog computer of the late 19th and early 20th centuries, constructed and set up to predict the ebb and flow of sea tides and the irregular variations in their heights – which change in mixtures of rhythms, that never (in the aggregate) repeat themselves exactly. Its purpose was to shorten the laborious and error-prone computations of tide-prediction. Such machines usually provided predictions valid from hour to hour and day to day for a year or more ahead. The first tide-predicting machine, designed and built in 1872–73, and followed by two larger machines on similar principles in 1876 and 1879, was conceived by Sir William Thomson. Thomson had introduced the method of harmonic analysis of tidal patterns in the 1860s and the first machine was designed by Thomson with the collaboration of Edward Roberts (1845-1933, assistant at the UK HM Nautical Almanac Office), and of Alexander Légé, who constructed it. In the US, another tide-predicting machine on a different pattern was designed by William Ferrel and built in 1881–82. Developments and improvements continued in the UK, US and Germany through the first half of the 20th century. The machines became widely used for constructing official tidal predictions for general marine navigation. They came to be regarded as of military strategic importance during World War I, and again during the Second World War, when the US No.2 Tide Predicting Machine, described below, was classified, along with the data that it produced, and used to predict tides for the D-Day Normandy landings and all the island landings in the Pacific War. Military interest in such machines continued even for some time afterwards. They were made obsolete by digital electronic computers that can be programmed to carry out similar computations, but the tide-predicting machines continued in use until the 1960s and 1970s. Several examples of tide-predicting machines remain on display as museum-pieces, occasionally put into operation for demonstration purposes, monuments to the mathematical and mechanical ingenuity of their creators.
Background
Modern scientific study of tides dates back to Isaac Newton's Principia of 1687, in which he applied the theory of gravitation to make a first approximation of the effects of the Moon and Sun on the Earth's tidal waters. The approximation developed by Newton and his successors of the next 90 years is known as the 'equilibrium theory' of tides. Beginning in 1776, Pierre-Simon Laplace made a fundamental advance on the equilibrium approximation, with the first description of oceanic tidal waters' dynamic responses to the tide-generating forces due to the Moon and Sun. Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by William Thomson's application of Fourier analysis to the tidal motions. Thomson's work in this field was then further developed and extended by George Darwin, the second son of Charles Darwin: George Darwin's work was based on the lunar theory current in his time. His symbols for the tidal harmonic constituents are still used. Darwin's harmonic developments of the tide-generating forces were later brought by A. T. Doodson up to date and extended in light of the new and more accurate lunar theory of E. W. Brown that remained current through most of the twentieth century. The state to which the science of tide-prediction had arrived by the 1870s can be summarized: Astronomical theories of the Moon and Sun had identified the frequencies and strengths of different components of the tide-generating force. Combining Moon and Sun forces was described by W. K. Clifford: "The use of the jointed parallelogram opqr for compounding harmonic motions of different periods is exemplified in Sir W. Thomson's Tidal Clock. The clock has two hands whose lengths are proportional to the solar and lunar tides respectively, while their periods of revolution are equal to the periods of those tides. A jointed parallelogram is constructed, having the hands of the clock for two sides. If the clock is properly set, the height of that extremity of the parallelogram which is furthest from the centre will be continually proportional to the height of the compound tide." But effective prediction at any given place called for measurement of an adequate sample of local tidal observations, to show the local tidal response at those different frequencies, in amplitude and phase. Those observations had then to be analyzed, to derive the coefficients and phase angles. Then, for purposes of prediction, those local tidal constants had to be recombined, each with a different component of the tide-generating forces to which it applied, and at each of a sequence of future dates and times, and then the different elements finally collected together to obtain their aggregate effects. In the age when calculations were done by hand and brain, with pencil and paper and tables, this was recognized as an immensely laborious and error-prone undertaking. Thomson recognized that what was needed was a convenient and preferably automated way to evaluate repeatedly the sum of tidal terms such as:
A 1 cos ( ω 1 t + ϕ 1 ) + A 2 cos ( ω 2 t + ϕ 2 ) + A 3 cos ( ω 3 t + ϕ 3 ) + … {\displaystyle A_{1}\cos(\omega _{1}t+\phi _{1})+A_{2}\cos(\omega _{2}t+\phi _{2})+A_{3}\cos(\omega _{3}t+\phi _{3})+\ldots }
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