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.

In Part I we saw how a geometric and trigonometric analysis of the Sun-Earth-Moon system leads to the following conclusion:

The vertical component of the combined solar and lunar ``tide-producing force'' at a point P on the Earth's surface is, as a function of time, a finite linear combination of sine and cosine terms sin(v_it), cos(v_jt), whose speeds v₁, v₂, v₃, ... are certain small integral linear combinations of the five astronomical frequencies T, h, s, p, N.

The idea behind the Fourier analysis of tides is that the height of the water at P should also be such a linear combination. In Laplace's words: ``The state of oscillation of a system of bodies in which the primitive conditions of movement have disappeared through friction is coperiodic with the forces acting on the system." (Mécanique Céleste, Book XIII Ch I, quoted in George Darwin's Tides entry, Encylopedia Brittanica, IX or XI edition.) The coefficients in that linear combination depend on the resonances between the speeds of the various terms and the natural frequencies of the bodies of water on which P abuts, in the style of classic harmonic excitation-reponse problems.

In Part II we saw how traditional Fourier analysis (which handles periodic functions) can be adapted to the non-periodic phenomenon of the tides. In particular, the tide at any port P can be written as a sum
```
H(t) = A₀ + A₁cos(v₁t) + B₁sin(v₁t) + A₂cos(v₂t) + B₂sin(v₂t) + ...
```
where the v_i are as above (only about 37 are ever used) and the coefficients A₀, A₁, B₁, A₂, B₂, ... are calculated from the tidal record at P.
In practice, the two terms A_icos(v_it) and B_isin(v_it) are amalgamated into a single term C_icos(v_it + p_i) using the trigonometric identity
C cos (vt + p) = C cos vt cos p - C sin vt sin p = A cos vt + B sin vt,
which leads to the equations C cos p = A, -C sin p = B with solution C = (A² + B²)^1/2, p = arctan(-B/A). The term
C_icos(v_it + p_i)
is called the constituent corresponding to the speed v_i; C_i is its amplitude; p_i is its phase.
In this part we will examine the remaining problem: how to go from the expression for H(t) to a prediction of the times and heights of the high and low tides at P on any given day.

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.

--Tony Phillips
Stony Brook