**IV. HARMONIC ANALYSIS**

22. Methods of Applying Theory to Practice

The comparison between tidal observations and tidal theories, and the formation of tables predicting the tidal oscillations of the sea, have been carried out in two different ways, which may be called the "synthetic" and the "analytic."

The semi-diurnal rise and fall of tide with the weekly alternation of spring and neap would naturally suggest to the investigator to make his formula conform to the apparent simplicity of the phenomenon. He would seek to represent the height of water by either one or two periodic functions with a variable amplitude; such a representation is the aim of the synthetic method. That method has been followed by all the great investigators of the past,—Newton, Bernoulli, Maclaurin, Laplace, Lubbock, Whelwell, Airy. Since at European ports the two tides which follow one another on any one day are nearly equal, or, in other words, there is scarcely a sensible diurnal tide, these investigators bestowed comparatively little attention to the diurnal tides. If these are neglected, the synthetic method is simple, for a single function suffices to represent the tide. In non-European ports, however, the diurnal tide is sometimes so large as to mask the semi-diurnal, and to make only a single instead of a double high water in twenty-four hours. To represent this diurnal tide in the synthetic method we are compelled to introduce at least one more function. There should also be a third function representing the tides of long period; but until the last few years these tides have scarcely been considered, and therefore we, shall have little to say of them in explaining the synthetic method. The expression for the tide-generating forces due to either sun or moon consists of three terms, involving the declinations and hour-angles of the planet. One of these terms for each goes through its period approximately twice a day, a second once a day, and the third varies slowly (§ 7). The mathematical basis of the synthetic method consists of a synthesis of the mathematical formulae. The semi-diurnal term for the moon is fused with that for the sun, and the same process is carried out for the diurnal and slowly varying terms.

A mass of tidal observation at a place where the diurnal tide is small, even if, as in all the older observations, it consists merely of heights and times of high and low water, soon shows that the fusion of two simple harmonic or periodic functions is insufficient to represent the state of tide; and the height and time of high water are found to need corrections for the variations of declination, of motion in right ascension, and of the parallaxes of both bodies.

But when continuous tide-gauges were set up far more extended data than those of the older observations became accessible to the investigator, and more and more corrections were found to be expedient to adapt the formulae to the facts. A systematic method of utilizing all the data became also a desideratum. This state of matters led Sir W. Thomson to suggest the analytic method.**[Footnote 363-1]** It is true that the dynamical foundations of that method have always met lain below the surface of the synthetic method, and have constantly been appealed to for the theoretical determination of corrections; nevertheless, we must regard the explicit adoption of the analytic method as a great advance. In this method we conceive the tidal forces or potential due to each disturbing body to be developed into a series of terms each consisting of a constant (determined by the elements of the planet’s orbit and the obliquity of the ecliptic) multiplied by a simple harmonic function of the time. Thus in place of the terms of the synthetic method for the three classes of tides we have an indefinitely long series of terms for each of the three classes. The loss of simplicity in the expression for the forces is far more than counterbalanced by the gain of facility for the discussion of the oscillations of the water. This facility arises from the great dynamical principle of forced oscillations, which we have explained in the historical sketch. Applying this principle, we see that each individual term of the harmonic development of the tide-generating forces corresponds to an oscillation of the sea of the same period, but the amplitude and phase of that oscillation must depend on a network of causes of almost inextricable complication. The analytic method, then, represents the tide at any port by a series of simple harmonic terms whose period is determined from theoretical considerations, but whose amplitude and phase are found from observation. Fortunately the series representing the tidal forces converges with sufficient rapidity to permit us to consider only a moderate number of harmonic terms in the series.

Now it seems likely that the corrections which have been applied in the use of the synthetic method might have been clothed in a more satisfactory and succinct mathematical form had investigators first carried out the harmonic development. In this article we shall therefore invert history and come back on the synthetic method from the analytic, and shall show how the formulae of correction stated in harmonic language may be made comparable with them in synthetic language. One explanation is expedient before proceeding with the harmonic development. There are certain terms in the tide-generating forces of the moon, depending on the longitude of the moon’s nodes, which complete their revolution in 18·6 years. Now it has been found practically convenient, in the application of the harmonic method, to follow the synthetic plan to the extent of classifying together terms whose speed differs only in consequence of the movement of the moon’s node, and at the same time to conceive that there is a small variability in the intensity of the generating forces.

**Footnote**

363-1 Airy and after him Chazallon, appear to have been amongst the first to use a kind of harmonic analysis for reducing tidal observations; but, as Airy did not emancipate himself from the use of hour-angles, declinations, &c., his work can hardly be considered as an example of the analytic method; see his "Tides and Waves," and Hatt’s *Phénomène des Marées*, Paris, 1885.

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