Fourier Analysis of Ocean Tides III
With JAVA animation by Bill Casselman
2. A conversation in a railway carriage
In 1872 Kelvin was preoccupied with the tidal prediction,
and in particular with the problem of summing a large
number of harmonic motions with irrationally
related frequencies. Here is how the solution came to him,
in his own words (Mathematical and Physical Papers,
Vol. VI Cambridge 1911, p. 286).
Dramatis Personae. The Author: William Thompson, later (1892) Lord
Kelvin. Mr Tower: Beauchamp Tower, an engineer and inventor. His
1883 experiments on lubrication were the basis for the ``Reynolds Equation,'' later treated by
Sommerfeld. Mr White: James White of
Glascow, ``Philosophical Instrument Maker to the University.''
The model was soon succeeded by the First Tide Predicting Machine,
which was completed in 1875, and could sum 10 tidal
constituents. It was succeeded in turn by a second (about 1880), a third
(about 1883) and a ``Fourth British Tide Predictor'' (1910). The machine
illustrated here is the third. Machines like this were constructed
as late as 1948, when the Doodson-Légé Tide Predicting Machine was built
in Liverpool, where it is still on display.
Click for larger image
A wire is fixed at the right and passes
alternately over and under 15 movable pulleys,
after which it suspends a weight
(in this image; in practice, an ink bottle with a pen). Each of the
movable pulleys is driven in a vertical simple harmonic motion, as
The crank also moves a strip of paper horizontally in front of the
pen (this is not shown) to record the predicted tidal curve. Image
from Kelvin, Mathematical and Physical Papers, Vol. VI, opposite
page 304. The planning of the gearing, which gives excellent
rational approximations to the astronnomical ratios, was carried
out by the mathematician Edward Roberts, who also supervised the
- Turning the crank it drives eleven gear
assemblies. The gear ratios are chosen so that the speeds of the
output gears in each assembly are uniformly proportional to the
speeds of the tidal constituents being summed. (Two of the assemblies,
#3 and #9 counting from the left, have two output gears, corresponding
to pairs of rationally related constituents; one of them, #11,
- Each output gear drives
one of the movable pulleys by an linkage which converts the
rotation of the gear into vertical harmonic motion. The linkage
attachment to the gear is by an eccentric pin mounted at an adjustable radius.
In this way the amplitude of the vertical motion can be set to match
the coefficient of the corresponding constituent at the port in
question. Likewise the initial angle of each pin can be set
so that all the constituents are in the correct phase for the
moment the simulation is to start.
- One last wrinkle: the pulleys
in the lower row must be set 180 degrees out of phase with those
in the upper, because when they go up, the pen goes down. The motion of
the pen is then exactly twice the sum of the motions of the pulleys.
The functioning of this mechanism, and the way in which the various
constituents contribute to the tide, are illustrated in a JAVA applet
Bill Casselman of the University of British Columbia.
Click on image to activate applet.