Tide-predicting machine

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.[8]

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.[8][9]

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.[8] 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:

${\displaystyle A_{1}\cos(\omega _{1}t+\phi _{1})+A_{2}\cos(\omega _{2}t+\phi _{2})+A_{3}\cos(\omega _{3}t+\phi _{3})+\ldots }$

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 ${\displaystyle t}$. 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

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 ${\displaystyle t}$, can then be expressed as ${\displaystyle A_{1}\cos(\omega _{1}t+\phi _{1})}$, where ${\displaystyle A_{1}}$ is the radial distance from the wheel's center to the peg, ${\displaystyle \omega _{1}}$ is the rate at which the wheel turns (in radians per unit of time), and ${\displaystyle \phi _{1}}$ 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

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.[10]

They can be seen in London,[24] Washington,[25] Liverpool,[26] and elsewhere, including the Deutsches Museum in Munich.

• Tide clock
• Tide table
• Tide-Predicting Machine No. 2

• T Ehret (2008), "Old Brass Brains - Mechanical Prediction of Tides", ACSM Bulletin, June 2008, pages 41–44.
• W Ferrel (1883), "A maxima and minima tide-predicting machine", in U S Coast Survey (1883), Appendix 10, pages 253-272.
• E G Fischer (1912), "The Coast and Geodetic Survey Tide Predicting Machine No. 2", Popular Astronomy, vol.20 (1912), pages 269-285.
• Institution of Civil Engineers (London), Proceedings volume 65 (1881), giving discussion after presentation of tide machines, minuted at pages 25–64.
• Parker, Bruce (September 2011). "The tide predictions for D-Day" (PDF). Physics Today. 64 (9): 35–40. doi:10.1063/PT.3.1257. Retrieved 27 July 2023.
• E Roberts (1879), "A new Tide-predicter", Proceedings of the Royal Society, xxix (1879), pages 198-201.
• Science (1884) [no author cited], "The Maxima and Minima Tide-Predicting Machine", Science, Vol.3 (1884), Issue 61, pp. 408–410.
• W Thomson (1881), "The tide gauge, tidal harmonic analyser, and tide predicter", Proceedings of the Institution of Civil Engineers, vol.65 (1881), pages 3–24.
• US Department of Commerce, Special Publication No.32 (1915), "Description of the US Coast and Geodetic Survey Tide-Predicting Machine No.2".
• P L Woodworth (2016), "An inventory of tide prediction machines", National Oceanography Centre Research and Consultancy Report No. 56.

Wikimedia Commons has media related to Tide-predicting machines.

• American Mathematical Society (2009), Fourier Analysis of Ocean Tides, II.2, showing effect of components at incommensurable frequencies.
• American Mathematical Society/Bill Casselman (2009), animated JAVA simulation based on Kelvin's Tide Predicting Machine (the animation shows computing 7 harmonic components).
• Deutsches Museum, Munich, online exhibit of 2nd German tide-predicting machine (described in English).
• German Maritime Museum (online exhibit of tide prediction and tide-predicting machines, in German).
• NOAA online exhibit of US No 2 Tide Predicting Machine with additional images.
• Norway online exhibit of Norwegian tide-predicting history & use of TPM (in English).
• National Oceanography Centre, Liverpool, exhibit of Roberts-Légé and Doodson-Légé Tide Predicting Machines.
• Science Museum, South Kensington, London: Kelvin Tide Predictor in the Science Museum London, along with a closer view.
• The Smithsonian National Museum of American History, Behring Center online exhibit The Ferrel tide predicting machine of 1881-2.