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"TIDES ( see 26.938). - The present century has seen a marked increase in the interest taken among foreign scientists in the study of the tides, while in Great Britain the subject again received much attention after the close of the World War.

1 Observation

2 General Distribution of Off-shore Tides

3 Dynamical Theory of the Tides

4 Harmonic Analysis

5 Atmospheric Pressure and Wind

6 Friction

7 History and Bibliography


The automatic tide gauges which are distributed along the coasts (in Great Britain very irregularly and chiefly according to the needs or caprices of harbour authorities) require much more attention than it has been the custom to pay to them. The errors in both elevation and time of their elevation-time graphs should be determined by independent observation at least once a day, as such errors very easily attain serious dimensions.

But the outstanding scientific need of the present time is for offshore observations. Not only do the great tidal movements of the ocean remain practically unobserved, but in the middle of the Irish Sea, for example, there is a discrepancy of 40 m. between the cotidal lines of different authoritative charts.

Off-shore elevations have been recorded by personal soundings (at the Dutch lightships, for example) but a number of attempts have been made to construct a self-registering gauge which, when placed on the bottom of the sea, will give a continuous pressure-time record. From such a record it is of course easy to pass to an elevationtime relation. Up to 1921 the gauge which appeared to have met with most success was that of M. Fa y e, of the French Marine, and even this had not worked in water of depth greater than zoo metres.

Continuous current observations are required at all depths.

A knowledge of currents is of immense importance both commercially and scientifically, and the effect of currents on mines during the World War caused much attention to be paid to them by naval authorities. Tidal currents are oscillatory, but observed currents have, as a rule, a residual drift which is of particular importance in general oceanographical or fishery research.

Surface currents have been measured by floating logs (as for most of the data published by the British Admiralty) but series of ob servations at frequent intervals, especially of currents below the surface, are usually made by current meters. One of the commonest of these - the Ekman meter - registers the mean speed and direction of the current during the interval of time it is in operation, the former by a small propeller actuating a revolution counting apparatus and the latter by a vane attached to an apparatus dropping shot into sectorial boxes on a compass card. It has thus to be hauled up to the surface for each reading. Continuous recording instruments are much needed and though some have been invented they do not appear to have been much used. Owing chiefly to the trouble of keeping a meter fixed relatively to the bottom, the accurate measurement of currents is a matter of great difficulty. Other data for residual currents or drifts are given by observations of weighted bottles or other forms of floating bodies, or by instruments so contrived as to float near the sea bottom.

General Distribution of Off-shore Tides

Much attention is now paid to the "amphidromic points," at which there is no rise and fall of the water and out from which the co-tidal lines radiate. Harris' charts of co-tidal lines contain a number of these points and so does the new chart of R. Sterneck (Sitzb. d. Akad. Wissensch., Wien, 129, 1920), which is based on all available data.

All recent charts of co-tidal lines for the North Sea agree in placing an amphidromic point in the southern region, and one of the services of the Favb gauge has been to give fresh observational verification of its existence (Comptes Rendus, 151, p. 803, 1910).

Dynamical Theory of the Tides

As regards the tidal dynamics of completely defined bodies of water, the only basins which had yielded to mathematical treatment up to 1914 were those of a flat circular sea, the depth of which was a function only of the distance from the centre, and an ocean covering the whole globe with the depth a function only of the latitude. The details for zonal basins of uniform depth have since been worked out by G. R. Goldsbrough (Proc. London Math. Soc., 14, 1914; 15, 1915).

Two attempts have been made, however, to bring some of the latest results of pure mathematics to bear on the general problem. In 1910 Poincare published his transformation of the dynamical equations from the differential to the integral form (Lecons de Mecanique Celeste, t. 3). The theory of integral equations has grown up almost entirely since 1900; its results are perfectly general and are stated explicitly in terms of direct operations. But in the case of tidal problems the arithmetical labour necessary to carry out these operations is so prodigious as to prove quite prohibitive even for the reproduction of known solutions: nevertheless, the theory is valuable for the establishment of existences.

Utilizing these existence-theorems J. Proudman (Proc. London Math. Soc., 18, 1917) has been able to specify the tidal state of an ocean by means of an infinite number of coordinates of the Lagrangian type, and then to transform the differential equations into an infinite set of linear algebraic equations. This has afforded a real prospect that the number of geometrically simple basins for which the tidal dynamics is completely known, may be increased.

The explanation, on dynamical principles, of the observed features of tides in small seas has been considerably advanced, chiefly by A. Defant and R. Sterneck. See Denkschr. d. Akad. Wissensch., Wien, 96 (1919). Sitzungsberichte, 123 (1914), 124 (1915), 129 (1902). The method of treatment only applies to elongated bodies of water and applications have been made to the Red Sea, the Persian Gulf, the English Channel, the Irish Sea and the Adriatic Sea. The motion is assumed to consist of a longitudinal oscillation sustained chiefly by the tides outside, with a transverse surface gradient sustained by the longitudinal current through the earth's rotation.

Other parts of the dynamical theory which have undergone development are those relating to slowly rotating seas and oceans, limiting forms of long period tides and the diffraction of tidal waves. See Rayleigh, Proc. Roy. Soc. (A) '82 (1909); J. Proudman, Proc. London Math. Soc., 12 (1913), 13 (1913), 14 (1914). In this connexion it may be mentioned that there is an erroneous statement in 26.957 §34, to the effect that the existence in the ocean of continental barriers would have the same effect as that attributed by Laplace to friction. In the actual oceans limiting forms of long period tides are possible which do not take the " equilibrium " values.

Harmonic Analysis

From 1883 up to the present time the standard harmonic development of the generating potential has been that of G. H. Darwin. Quite recently A. T. Dood son has made a new development, working to a much higher order of approximation than Darwin, and has found that there is a very large number of other constituents which, while certainly being smaller than those of Darwin, are not very much smaller, and in their aggregate may be important. In other words, the convergence of the series of constituents is not so rapid as has been assumed.

A corresponding state of affairs exists with regard to over tides and compound tides. For certain British stations A. T. Doodson, being led by dynamical principles, has found it possible to obtain a practically complete representation of the quarter diurnal tides, but it involves many more harmonic constituents than have ever been sought for by the customary methods. This representation is susceptible of very simple algebraic statement and numerical application but cannot be used on the existing predicting machines.

The present state of analysis is not. satisfactory. The harmonic constants do not represent completely the records analysed: for certain British stations the discrepancy may have a quarter-diurnal range of one foot and a semidiurnal range of one foot.

J. Proudman (British Assoc. Report, 1920, p. 323) has given an account of British work on harmonic analysis with a bibliography and lists of analyses made.

Tide Tables cannot be regarded as satisfactory even for such practical purposes as docking large vessels or navigating over shallows, while for a hopeful study of meteorological effects they are almost useless. The main deficiency appears to be one of analysis of records; for others, see A. T. Doodson, Brit. Assoc. Report, 1920, p. 321.

When the astronomical tides can be predicted with the same degree of accuracy as the resultant tides can be observed, there appears to be no reason why short date predictions of meteorological tides - obviously of great importance - should not be attempted.

Atmospheric Pressure and Wind

The effects of meteorological influences on the tides have been much studied, especially by the Scandinavians. As regards the relative importance of atmospheric pressure and wind, a general conclusion appears to be that at a station in the immediate neighbourhood of a wide expanse of deep ocean, the direct pressure effect predominates, whereas at a station in a landlocked and shallow sea, the wind effect predominates. The detailed study of these effects is rendered very difficult by the uncertainties in the predictions of astronomical tides, and most investigations have dealt with mean effects over long intervals of time. There is much literature on the subject: see for example D. la Cour, Danske Meteorologiske Institut, Meddelelser, 1 (1913), 4 (1917)

R. Witting, Fennia 39, 5 (Helsingfors 1918). The 1917 memoir of la Cour is a detailed study of the effects of a storm.


If tidal motion were everywhere non-turbulent then the amount of friction in the oceans would be quite insufficient to account for the outstanding discrepancy between theory and observation in the motion of the moon. See R. O. Street, Proc. Roy. Soc.

(A) 93 (1917). But the motion associated with large tides in shallow seas is undoubtedly turbulent and though this has long been recognized it is only recently that numerical estimates of its amount have been made. See G. I. Taylor, Phil. Trans. (A), 220 (1920).

H. Jeffreys, ibid 221 (1920), concludes that the total amount of friction is just about sufficient to account for the discrepancy mentioned. He takes the chief contributing areas to be Bering Sea, the Yellow Sea, Malacca Strait and the American N.W. Passage.

History and Bibliography

To the list of outstanding names in the history of the theory of the tides should be added those of G. H. Darwin and H. Lamb. The chief contributions of the former were his elaboration of the methods of harmonic analysis and his farreaching cosmogonical deductions as to the consequences of tidal friction. The chief contribution of the latter is in connexion with steady motions and the discrimination of free oscillations in the general dynamical theory. Additions to the list of books on tides are R. A. Harris, Manual of Tides v. (1907); H. Poincare, Lecons de Mecanique Celeste, t. 3 (1910) and O. Kriimmel, Handbuch der Oceanographic, B. 2, C. 3 (1911). (J. P.*)

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Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Tides'. 1911 Encyclopedia Britanica. 1910.

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