Fourier Analysis of Ocean Tides III
With JAVA animation by Bill Casselman
Additional Web resources on tides in general: The NOAA Website presents "Our Restless Tides," an explanation of the causes of tides, by the National Ocean Service. Science@NASA's Ocean Tides Lost and Found (June 15, 2000) has among other things a remarkable animation of the world-wide tide, made from sea-level measurements by the TOPEX/Poseidon satellite altimeter. The Bigelow Laboratory for Ocean Sciences has a tide page with many useful links. On other worlds: Space Science News for March 8, 2000 published "Terrible Tides" about tidal forces on Jupiter's moons.
Images and explanations of tide predicting machines are given on the NOAA-NOS CO-OPS Tide Predicting machines site. Besides Kelvin's own writings, a reference for his work in developing mechanical devices is George Green and John T. Lloyd, Kelvin's Instruments and the Kelvin Museum, University of Glascow, 1970 (which also contains a wonderful evocation, by Green, of Kelvin's lecturing style). My main reference on tidal theory and analysis is Paul Schureman, Manual of Harmonic Analysis and Prediction of Tides, United States Government Printing Office 1958.
1. Setting up the problem
This is the last part of a 3-part series on the topic.
This problem sounds simple. Every Freshman learns that to find maxima or minima of a function H(t), you calculate its derivative H'(t) and solve the equation H'(t) = 0. But when H(t) is a linear combination of irrationally related sines and cosines, as is the case for the tide, that last equation cannot be solved by analytic methods. Solutions can be pinned down by a series of better and better approximations, but the whole procedure must be repeated for each solution.
The amount of calculation involved in solving such an equation drove Kelvin to look for a mechanical way of summing a large number of harmonic functions. It is easy enough to create separate periodic motions with given amplitudes and frequencies. If they could be summed, then one could organize a high-speed simulation of the function H(t). A month's worth of tides could be simulated in an hour; the data for a tide table (usually the time and height of the high tides) could be recorded and published.
Historical note: It is clear from the dates that the impetus behind the British Association's investigation of the tides was the need for reliable information for ports in India. There is an almost exact 19-year cycle in the joint pattern of equinoxes and solstices, and phases of the moon, so the tidal record repeats almost exactly if you wait long enough: this fact had been used to prepare useful tables for European ports. But when the British took over Bombay and Calcutta, they could not wait nineteen years for accurate tide predictions.
In the rest of this column we will examine Kelvin's solution to this problem, his Tide Predicting Machine, and its mechanical descendants.
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