Tide-predicting machine

Last updated
10-component tide-predicting machine of 1872-3, conceived by Sir William Thomson (Lord Kelvin), and designed by Thomson and collaborators, at the Science Museum, South Kensington, London DSCN1739-thomson-tide-machine.jpg
10-component tide-predicting machine of 1872-3, conceived by Sir William Thomson (Lord Kelvin), and designed by Thomson and collaborators, at the Science Museum, South Kensington, London

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. [1] 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 (assistant at the UK HM Nautical Almanac Office), and of Alexander Légé, who constructed it. [2]

In the US, another tide-predicting machine on a different pattern was designed by William Ferrel and built in 1881–2. [3] 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, [4] 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. [5] Military interest in such machines continued even for some time afterwards. [6] 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. [7]

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.


William Ferrel's tide-predicting machine of 1881-2, now at the Smithsonian National Museum of American History 099-ferreltpm.jpg
William Ferrel's tide-predicting machine of 1881-2, now at the Smithsonian National Museum of American History

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 the 1770s, Pierre-Simon Laplace made a fundamental advance on the equilibrium approximation by bringing into consideration non-equilibrium dynamical aspects of the motion of tidal waters that occurs in response 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. 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:

containing 10, 20 or even more trigonometrical terms, so that the computation could conveniently be repeated in full for each of a very large number of different chosen values of the date/time . This was the core of the problem solved by the tide-predicting machines.


Thomson conceived his aim as to construct a mechanism that would evaluate this trigonometrical sum physically, e.g. as the vertical position of a pen that could then plot a curve on a moving band of paper.

mechanism for generating sinusoidal motion component 099-tmpshaft.jpg
mechanism for generating sinusoidal motion component

There were several mechanisms available to him for converting rotary motion into sinusoidal motion. One of them is shown in the schematic (right). A rotating drive-wheel is fitted with an off-center peg. A shaft with a horizontally-slotted section is free to move vertically up and down. The wheel's off-center peg is located in the slot. As a result, when the peg moves around with the wheel, it can make the shaft move up and down within limits. This arrangement shows that when the drive-wheel rotates uniformly, say clockwise, the shaft moves sinusoidally up and down. The vertical position of the center of the slot, at any time , can then be expressed as , where is the radial distance from the wheel's center to the peg, is the rate at which the wheel turns (in radians per unit of time), and is the starting phase angle of the peg, measured in radians from the 12 o'clock position to the angular position where the peg was at time zero.

This arrangement makes a physical analog of just one trigonometrical term. Thomson needed to construct a physical sum of many such terms.

At first he inclined to use gears. Then he discussed the problem with engineer Beauchamp Tower before the British Association meeting in 1872, and Tower suggested the use of a device that (as he remembered) was once used by Wheatstone. It was a chain running alternately over and under a sequence of pulleys on movable shafts. The chain was fixed at one end, and the other (free) end was weighted to keep it taut. As each shaft moved up or down it would take up or release a corresponding length of the chain. The movements in position of the free (movable) end of the chain represented the sum of the movements of the different shafts. The movable end was kept taut, and fitted with a pen and a moving band of paper on which the pen plotted a tidal curve. In some designs, the movable end of the line was connected instead to a dial and scale from which tidal heights could be read off.

Thomson's design for the third tide-predicting machine, 1879-81 099-tpm3-sk.jpg
Thomson's design for the third tide-predicting machine, 1879-81

One of Thomson's designs for the calculating part of a tide-predicting machine is shown in the figure (right), closely similar to the third machine of 1879-81. A long cord, with one end held fixed, passed vertically upwards and over a first upper pulley, then vertically downwards and under the next, and so on. These pulleys were all moved up and down by cranks, and each pulley took in or let out cord according to the direction in which it moved. These cranks were all moved by trains of wheels gearing into the wheels fixed on a drive shaft. The greatest number of teeth on any wheel was 802 engaging with another of 423. All the other wheels had comparatively small numbers of teeth. A flywheel of great inertia enabled the operator to turn the machine fast, without jerking the pulleys, and so to run off a year's curve in about twenty-five minutes. The machine shown in the figure was arranged for fifteen constituents in all.

Thomson acknowledged that the use of an over-and-under arrangement of the flexible line that summed the motion components was suggested to him in August 1872 by engineer Beauchamp Tower. [8]


The Kelvin machine

The first tide predicting machine (TPM), designed in 1872 and of which a model was exhibited at the British Association meeting in 1873 [9] (for computing 8 tidal components), followed in 1875-6 by a machine on a slightly larger scale (for computing 10 tidal components), was designed by Sir William Thomson (who later became Lord Kelvin). [10] The 10-component machine and results obtained from it were shown at the Paris Exhibition in 1878.

Thomson was also responsible for originating the method of harmonic tidal analysis, and for devising a harmonic analyzer machine, which partly mechanized the evaluation of the constants from the gauge readings.

The Roberts machine

An enlarged and improved version of the machine, for computing 20 tidal components, was built for the Government of India in 1879, and then modified in 1881 to extend it to compute 24 harmonic components. [11] British Tide Predictor No.2, after initial use to generate data for Indian ports, was used for tide prediction for the British empire beyond India, and transferred to the National Physical Laboratory in 1903. British Tide Predictor No.3 was sold to the French Government in 1900 and used to generate French tide tables.

In these machines, the prediction was delivered in the form of a continuous graphical pen-plot of tidal height against time. The plot was marked with hour- and noon-marks, and was made by the machine on a moving band of paper as the mechanism was turned. A year's tidal predictions for a given place, usually a chosen seaport, could be plotted by the 1876 and 1879 machines in about four hours (but the drives had to be rewound during that time).

Ferrel machine, US Tide Predicting Machine No. 1

In 1881–2, another tide predicting machine, operating quite differently, was designed by William Ferrel and built in Washington under Ferrel's direction by E. G. Fischer (who later designed the successor machine described below, which was in operation at the US Coast and Geodetic Survey from 1912 until the 1960s). [12] Ferrel's machine delivered predictions by telling the times and heights of successive high and low waters, shown by pointer-readings on dials and scales. These were read by an operator who copied the readings on to forms, to be sent to the printer of the US tide-tables.

These machines had to be set with local tidal constants special to the place for which predictions were to be made. Such numbers express the local tidal response to individual components of the global tide-generating potential, at different frequencies. This local response, shown in the timing and the height of tidal contributions at different frequencies, is a result of local and regional features of the coasts and sea-bed. The tidal constants are usually evaluated from local histories of tide-gauge observations, by harmonic analysis based on the principal tide-generating frequencies as shown by the global theory of tides and the underlying lunar theory.

Development and improvement based on the experience of these early machines continued through the first half of the 20th century.

Tide-Predicting Machine No. 2 ("Old Brass Brains"). The operator powered the machine by turning the crank at the left. The machine stopped when the simulation reached high and low tides, at which time the operator recorded the tide height and the day and time from the dials on the machine's face. The tide curve drawn on the paper above the dials was retained in case questions were raised later about the calculations. Tide Predicting Machine No. 2 ("Old Brass Brains").jpg
Tide-Predicting Machine No. 2 ("Old Brass Brains"). The operator powered the machine by turning the crank at the left. The machine stopped when the simulation reached high and low tides, at which time the operator recorded the tide height and the day and time from the dials on the machine's face. The tide curve drawn on the paper above the dials was retained in case questions were raised later about the calculations.

US Tide Predicting Machine No. 2

US Tide Predicting Machine No. 2 ("Old Brass Brains") [13] was designed in the 1890s, completed and brought into service in 1912, used for several decades including during the second World War, and retired in the 1960s.

20th century

Tide-predicting machines were built in Germany during World War I, and again in the period 1935-8. [14]

Three of the last to be built were:

Excluding small portable machines, a total of 33 tide-predicting machines are known to have been built, of which 2 have been destroyed and 4 are presently lost. [17]

Display and demonstration

They can be seen in London, [18] Washington, [19] Liverpool, [20] and elsewhere, including the Deutsches Museum in Munich.


An online demonstration is available to show the principle of operation of a 7-component version of a tide-predicting machine otherwise like Thomson's (Kelvin's) original design. [21] The animation shows part of the operation of the machine: the motions of several pulleys can be seen, each moving up and down to simulate one of the tidal frequencies; and the animation also shows how these sinusoidal motions were generated by wheel rotations and how they were combined to form the resulting tidal curve. Not shown in the animation is the way in which the individual motions were generated in the machine at the correct relative frequencies, by gearing in the correct ratios, or how the amplitudes and starting phase angles for each motion were set in an adjustable way. These amplitudes and starting phase angles represented the local tidal constants, separately reset, and different for each place for which predictions were to be made. Also, in the real Thomson machines, to save on motion and wear of the other parts, the shaft and pulley with the largest expected motion (for the M2 tide component at twice per lunar day) was mounted nearest to the pen, and the shaft and pulley representing the smallest component was at the other end, nearest to the point of fixing of the flexible cord or chain, to minimize unnecessary motion in the most part of the flexible cord.

See also

Notes and references

  1. See American Mathematical Society (2009) II.2, showing how combinations of waves in non-commensurable frequencies cannot repeat their resultant patterns exactly.
  2. The Proceedings of the Inst.C.E. (1881) contains minutes of a somewhat disputatious discussion that took place in 1881 over who had contributed what details. Thomson acknowledged previous work of the 1840s relating to the general mechanical solution of equations, plus a specific suggestion he had from Beauchamp Tower to use a device of pulleys and a chain once used by Wheatstone; Thomson also credited Roberts with calculating the astronomical ratios embodied in the machine, and Légé with design of the drive gear details; Roberts claimed further credit for selecting other parts of the mechanical design.
  3. Ferrel (1883).
  4. During World War I, Germany built its first tide-predicting machine in 1915-16 when it could no longer obtain British hydrographic data (see Deutsches Museum exhibit, online), and when it specially needed accurate and independently-sourced tide data for conducting the U-boat campaign (see German Maritime Museum exhibit, online).
  5. See Ehret (2008) at page 44).
  6. During the 'cold war', East Germany built its own tide-predicting machine in 1953-5 "at unbelievable expense", see German Maritime Museum (online exhibit).
  7. The US No.2 machine was retired in the 1960s, see Ehret (2008); the machine used in Norway continued in use until the 1970s (see Norway online exhibit).
  8. Beauchamp Tower was initially referred to in Thomson's acknowledgements only as 'Mr Tower', but he was more fully identified in discussion between Thomson and E Roberts at the Institution of Civil Engineers (reported in the ICE minutes in the Proceedings, 1881).
  9. See Proceedings of the Inst.C.E. (1881), at page 31.
  10. see W Thomson (1881), a paper of Thomson's presented to the Institution of Civil Engineers in January 1881. Subsequent discussion at the same meeting of the Institution of Civil Engineers covered questions of history and priority about aspects of the design since 1872, see Proceedings for January 1881 especially pages 30-31. The design had been described at the British Association Meeting of 1872 and a model for an 8-component prototype shown at the British Association meeting of 1873.
  11. The 20-component instrument was described by E Roberts (1879).
  12. W Ferrel (1883); also E G Fischer (1912), at pages 273-275; also Science (1884).
  13. See Ehret, 2008 for its later history, and for its construction E G Fischer, and (1915) Description of the US Tide Predicting Machine No 2, see also NOAA.
  14. See German Maritime Museum online exhibit, and Deutsches Museum online exhibit.
  15. Norwegian Hydrographic Service - history.
  16. See German Maritime Museum (online exhibit).
  17. See P. L. Woodworth (2016): An inventory of tide prediction machines. National Oceanography Centre Research and Consultancy Report No. 56.
  18. The first complete tide-predicting machine, from 1872-3, by Thomson with contributions from Tower, Roberts, and Légé, is at the Science Museum, South Kensington, London.
  19. The first US tide-predicting machine by Ferrel, 1881-2, is exhibited at The Smithsonian National Museum of American History; and the second US tide-predicting machine, which gained the nickname "Old Brass Brains" (see Ehret, 2008), is exhibited at the NOAA offices in Silver Spring, MD (NOAA is the National Oceanographic and Atmospheric Administration).
  20. The Roberts-Légé and Doodson-Légé machines are exhibited in the Tide and Time exhibition at the Proudman Oceanographic Laboratory, Liverpool, UK.
  21. See American Mathematical Society/Bill Casselman (2009), animated JAVA simulation based on Kelvin's Tide Predicting Machine (the animation shows computing 7 harmonic components).


Related Research Articles

Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.

<span class="mw-page-title-main">Analog computer</span> Computer that uses continuously variable technology

An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved. In contrast, digital computers represent varying quantities symbolically and by discrete values of both time and amplitude.

Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic Analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience.

<span class="mw-page-title-main">Tide</span> Rise and fall of the sea level under astronomical gravitational influences

Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon and are also caused by the Earth and Moon orbiting one another.

In electrical engineering, the power factor of an AC power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit. Real power is the average of the instantaneous product of voltage and current and represents the capacity of the electricity for performing work. Apparent power is the product of RMS current and voltage. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power may be greater than the real power, so more current flows in the circuit than would be required to transfer real power alone. A power factor magnitude of less than one indicates the voltage and current are not in phase, reducing the average product of the two. A negative power factor occurs when the device generates real power, which then flows back towards the source.

<span class="mw-page-title-main">Internal wave</span> Type of gravity waves that oscillate within a fluid medium

Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance, the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid.

Joseph Proudman, CBE, FRS was a distinguished British mathematician and oceanographer of international repute. His theoretical studies into the oceanic tides not only "solved practically all the remaining tidal problems which are soluble within the framework of classical hydrodynamics and analytical mathematics" but laid the basis of a tidal prediction service developed with Arthur Doodson of great international importance.

<span class="mw-page-title-main">Beat (acoustics)</span> Term in acoustics

In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies.

<span class="mw-page-title-main">Tide table</span> Tabulated data used for tidal prediction

Tide tables, sometimes called tide charts, are used for tidal prediction and show the daily times and levels of high and low tides, usually for a particular location. Tide heights at intermediate times can be approximated by using the rule of twelfths or more accurately calculated by using a published tidal curve for the location. Tide levels are typically given relative to a low-water vertical datum, e.g. the mean lower low water (MLLW) datum in the US.

Condition monitoring is the process of monitoring a parameter of condition in machinery, in order to identify a significant change which is indicative of a developing fault. It is a major component of predictive maintenance. The use of condition monitoring allows maintenance to be scheduled, or other actions to be taken to prevent consequential damages and avoid its consequences. Condition monitoring has a unique benefit in that conditions that would shorten normal lifespan can be addressed before they develop into a major failure. Condition monitoring techniques are normally used on rotating equipment, auxiliary systems and other machinery like belt-driven equipment,, while periodic inspection using non-destructive testing (NDT) techniques and fit for service (FFS) evaluation are used for static plant equipment such as steam boilers, piping and heat exchangers.

Ripple in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

Atmospheric tides are global-scale periodic oscillations of the atmosphere. In many ways they are analogous to ocean tides. Atmospheric tides can be excited by:

<span class="mw-page-title-main">Theory of tides</span> Scientific interpretation of tidal forces

The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans under the gravitational loading of another astronomical body or bodies.

<span class="mw-page-title-main">Least-squares spectral analysis</span> Periodicity computation method

Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum based on a least-squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such problems. Unlike in Fourier analysis, data need not be equally spaced to use LSSA.

Arthur Thomas Doodson was a British oceanographer.

<span class="mw-page-title-main">Harmonic damper</span>

A harmonic damper is a device fitted to the free end of the crankshaft of an internal combustion engine to counter torsional and resonance vibrations from the crankshaft. This device must be interference fit to the crankshaft in order to operate in an effective manner. An interference fit ensures the device moves in perfect step with the crankshaft. It is essential on engines with long crankshafts and V8 engines with cross plane cranks, or V6 and straight-three engines with uneven firing order. Harmonics and torsional vibrations can greatly reduce crankshaft life, or cause instantaneous failure if the crankshaft runs at or through an amplified resonance. Dampers are designed with a specific weight (mass) and diameter, which are dependent on the damping material/method used, to reduce mechanical Q factor, or damp, crankshaft resonances.

Alberto dos Santos Franco the Admiral Franco was an officer of the Navy of Brazil. In his career, reached the rank of rear admiral

<span class="mw-page-title-main">Ball-and-disk integrator</span> Component used in mechanical computers

The ball-and-disk integrator is a key component of many advanced mechanical computers. Through simple mechanical means, it performs continual integration of the value of an input. Typical uses were the measurement of area or volume of material in industrial settings, range-keeping systems on ships, and tachometric bombsights. The addition of the torque amplifier by Vannevar Bush led to the differential analysers of the 1930s and 1940s.

<span class="mw-page-title-main">Tide-Predicting Machine No. 2</span> Mechanical analog computer

Tide-Predicting Machine No. 2, also known as Old Brass Brains, was a special-purpose mechanical computer that uses gears, pulleys, chains, and other mechanical components to compute the height and time of high and low tides for specific locations. The machine can perform tide calculations much faster than a person could do with pencil and paper. The U.S. Coast and Geodetic Survey put the machine into operation in 1910. It was used until 1965, when it was replaced by an electronic computer.

Nonlinear tides are generated by hydrodynamic distortions of tides. A tidal wave is said to be nonlinear when its shape deviates from a pure sinusoidal wave. In mathematical terms, the wave owes its nonlinearity due to the nonlinear advection and frictional terms in the governing equations. These become more important in shallow-water regions such as in estuaries. Nonlinear tides are studied in the fields of coastal morphodynamics, coastal engineering and physical oceanography. The nonlinearity of tides has important implications for the transport of sediment.