Encyclopædia Britannica, Ninth Edition/Tides/Chapter 1
I. On the Nature of Tides.
§1. Definition of Tide.
WHEN, as occasionally happens, a ship in the open sea meets a short succession of waves of very unusual magnitude, we hear of tidal waves; and the large wave caused by an earthquake is commonly so described. The use of the term " tide " in this connexion is certainly incor rect, but it has perhaps been fostered by the fact that such waves impress their records on automatic tide-gauges, as, for example, when the wave due to the volcanic outbreak at Krakatoa was thus distinctly traceable in South Africa, and perhaps even faintly at Brest. We can only adequately define a tide by reference to the cause which produces it. A tide then is a rise and fall of the water of the sea pro duced by the attraction of the sun and moon. A rise and fall of the sea produced by a regular alternation of day and night breezes, by regular rainfall and evaporation, or by any influence which the moon may have on the weather cannot strictly be called a tide. Such alternations may, it is true, be inextricably involved with the rise and fall of the true astronomical tide, but we shall here distinguish them as meteorological tides. These movements are the result of the action of the sun, as a radiating body, on the earth. Tides in the atmosphere would be shown by a regular rise and fall in the barometer, but such tides are undoubtedly very minute, and we shall not discuss them in this article, merely referring the reader to the Mécanique Céleste of Laplace, bks. i. and xiii. There are, however, very strongly marked diurnal and semi-diurnal inequalities of the barometer due to atmospheric meteorological tides. Sir William Thomson in an interesting speculation  shows that the interaction of these quasi-tides with the sun is that of a thermodynamic engine, whereby there is caused a minute secular acceleration of the earth's rotation. This matter is, however, beyond the scope of the present article. We shall here extend the term " tide " to denote an elastic or viscous periodic deformation of a solid or viscous globe under the action of tide-generating forces. In the techni cal part of the article by the term "a simple tide" we shall denote a spherical harmonic deformation of the water on the surface of the globe, or of the solid globe itself, multiplied by a simple harmonic function of the time.
§2. General Description of Tidal Phenomena.
If we live by the sea or on an estuary, we see that the retarded water rises and falls nearly twice a day; speaking more 50m p er exactly, the average interval from high water to high diem. water is about 12h 25m, so that the average retardation from day to day is about 50m. The times of high water are then found to bear an intimate relation with the moon's position. Thus at Ipswich high water occurs when the moon is nearly south, at London Bridge when it is south west, and at Bristol when it is east-south-east. For a very rough determination of the time of high water it is sufficient to add the solar time of high water on the days of new and full moon (called the "establishment of the port") to the time of the moon's passage over the meridian, either visibly above or invisibly below the horizon. The interval between the moon's passage over the meridian and high water varies sensibly with the moon's age. From new moon to first quarter, and from full moon to third quarter (or rather from and to a day later than each of these phases), the interval diminishes from its average to a minimum, and then increases again to the average; and in the other two quarters it increases from the average to a maximum, and then diminishes again to the average.
The range of the rise and fall of water is also subject to great variability. On the days after new and full moon the range of tide is at its maximum, and on the day after the first and third quarter at its minimum. The maximum is called "spring tide" and the minimum "neap tide," and the range of spring tide is usually between two and three times as great as that of neap tide. At many ports, how ever, especially non-European ones, two successive high waters are of unequal heights, and the law of variability of the difference is somewhat complex; a statement of that law will be easier when we come to consider tidal theories. In considering any tide we find, especially in estuaries, that the interval from high to low water is longer than that from low to high water, and the difference between the intervals is greater at spring than at neap.
 In other cases where the differ ence between the periods of rising and falling is consider able, there are, in each high water, two or three rises and falls. A double high water exists at Southampton.In a river the current continues to run up stream for some considerable time after high water is attained and to run down similarly after low water. Much confusion has been occasioned by the indiscriminate use of the term "tide" to denote a tidal current and a rise of water, and it has often been incorrectly inferred that high water must have been attained at the moment of cessation of the upward current. The distinction between "rising and falling" and "flowing and ebbing" must be carefully maintained in rivers, whilst it vanishes at the seaboard. If we examine the progress of the tide-wave up a river, we find that high water occurs at the sea earlier than higher up. If, for instance, on a certain day it is high water at Margate at noon, it is high water at Gravesend at a quarter past two, and at London Bridge a few minutes before three. The interval from low to high water diminishes also as we go up the river; and at some distance up certain rivers—as, for example, the Severn—the rising water spreads over the flat sands in a roaring surf and travels up the river almost like a wall of water. This kind of sudden rise is called a "bore."
When an estuary contracts considerably, the range of tide becomes largely magnified as it narrows; for example, at the entrance of the Bristol Channel the range of spring tides is about 18 feet, and at Chepstow about 50 feet.This augmentation of the height of the tide-wave is due to the concentration of the energy of motion of a large mass of water into a narrow space. At oceanic ports the tidal phenomena are much less marked, the range of tide being usually only 2 or 3 feet, and the interval from high to low water sensibly equal to that from low to high water. The changes from spring to neap tide and the relation of the time of high water to the moon's transit remain, how ever, the same as in the case of the river tides.
In long and narrow seas, such as the English Channel, the tide in mid-channel follows the same law as at a station near the mouth of a river, rising and falling in equal times; the current runs in the direction analogous to up stream for three hours before and after high water, and down stream for the same period before and after low water. But near the sides of channels and near the mouths of bays the changes of the currents are very complex; and near the headlands separating two bays there is usually at certain times a very swift current, termed a "race."
In inland seas, such as the Mediterranean, the tides are nearly insensible except at the ends of long bays. Thus at Malta the tides are not noticed by the ordinary observer, whilst at Venice they are conspicuous.
 that the water has been known to depart from its predicted height at London by as much as 5 feet. The effect of wind will certainly be different at each port. The discrepancy of opinion on this subject appears to be great, so much so that we hear of some observers concluding that the effect of the wind is insensible. Variations in barometric pressure also cause departures from the predicted height of water, high barometer corresponding to decrease of height of water. Roughly speaking, an inch of the mercury column will correspond to something less than a foot of water, but the effect seems to vary much at different ports.The effect of a strong wind on the height of tide is generally supposed to be very marked, especially in estu aries. In the case of an exceptional gale, when the wind veered round appropriately, Airy states
§3. General Explanation of the Cause of Tides.
The moon attracts every particle of the earth and ocean, and by the law of gravitation the force acting on any particle is directed towards the moon's centre, and is jointly proportional to the masses of the particle and of the moon, and inversely proportional to the square of the distance between the particle and the moon's centre. If we imagine the earth and ocean subdivided into a number of small portions or particles of equal mass, then the average, both as to direction and intensity, of the forces acting on these particles is equal to the force acting on that particle which is at the earth's centre. For there is symmetry about the line joining the centres of the two bodies, and, if we divide the earth into two portions by an ideal spherical surface passing through the earth's centre and having its centre at the moon, the portion remote from the moon is a little larger than the portion towards the moon, but the nearer portion is under the action of forces which are a little stronger than those acting on the further portion, and the resultant of the weaker forces on the larger portion is exactly equal to the resultant of the stronger forces on the smaller. If every particle of the earth and ocean were being urged by equal and parallel forces, there would be no cause for relative motion between the ocean and the earth. Hence it is the departure of the force acting on any particle from the average which constitutes the tidegenerating force. Now it is obvious that on the side of the earth towards the moon the departure from the average is a small force directed towards the moon; and on the side of the earth away from the moon the departure is a small force directed away from the moon. Also these two departures are very nearly equal to one another, that on the near side being so little greater than that on the other that we may neglect the excess. All round the sides of the earth along a great circle perpendicular to the line joining the moon and earth, the departure is a force directed inwards towards the earth's centre. Thus we see that the tidal forces tend to pull the water towards and away from the moon, and to depress the water at right angles to that direction. If we could neglect the rotations of the bodies, and could consider the system as at rest, we should find that the water was in equilibrium when elongated into a prolate ellipsoid with its long axis directed towards and away from the moon.
But it must not be assumed that this would be the case when there is motion. For, suppose that the ocean consisted of a canal round the equator, and that an earthquake or any other cause were to generate a great wave in the canal, this wave would travel along it with a velocity dependent on the depth. If the canal were about 13 miles deep, the velocity of the wave would be about 1000 miles an hour, and with depth about equal to the depth of our seas the velocity of the wave would be about half as great. We may conceive the moon's tide-generating force as making a wave in the canal and continually outstripping the wave it generates, for the moon travels along the equator at the rate of about 1000 miles an hour, and the sea is less than 13 miles deep. The resultant oscillation of the ocean must therefore be the summation of a series of partial waves generated at each instant by the moon and always falling behind her, and the aggregate wave, being the same at each instant, must travel 1000 miles an hour so as to keep up with the moon.
Now it is a general law of frictionless oscillation that, if a slowly varying periodic force acts on a system which would oscillate quickly if left to itself, the maximum ex cursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction of the excursion; but, if a quickly varying periodic force acts on a system which would oscillate slowly if left to itself, the maximum excursion on one side of the equili brium position occurs simultaneously with the maximum force in the direction opposite to that of the excursion. An example of the first is a ball hanging by a short string, which we push slowly to and fro; the ball will never quit contact with the hand, and will agree with its excursions. If, however, the ball is hanging by a long string we can play at battledore and shuttlecock with it, and it always meets our blows. The latter is the analogue of the tides, for a free wave in our shallow canal goes slowly, whilst the moon's tide-generating action goes quickly.Hence, when the system is left to settle into steady oscillation, it is low water under and opposite to the moon, whilst the forces are such as to make it high water at those times.
If we consider the moon as revolving round the earth, the water assumes nearly the shape of an oblate spheroid with the minor axis pointed to the moon. The rotation of the earth in the actual case introduces a complexity which it is not easy to unravel by general reasoning. We can see, however, that if water moves from a lower to a higher latitude it arrives at the higher latitude with more velocity from west to east than is appropriate to its lati tude, and it will move accordingly on the earth's surface. Following out this conception, we see that an oscillation of the water to and fro between south and north must be accompanied by an eddy. Laplace's solution of the diffi cult problem involved in working out this idea will be given below.
The conclusion at which we have arrived about the tides of an equatorial canal is probably more nearly true of the tides of a globe partially covered with land than if we were to suppose the ocean at each moment to assume the prolate figure of equilibrium. In fact, observation shows that it is more nearly low water than high water when the moon is on the meridian. If we consider how the oscillation of the water would appear to an observer carried round with the earth, we see that he will have low water twice in the lunar day, somewhere about the time when the moon is on the meridian, either above or below the horizon, and high water half way between the low waters.
the range of spring tide will be about three times that of neap tide.If the sun be now introduced, we have another similar tide of about half the height, and this depends on solar time, giving low water somewhere about noon and midnight. The superposition of the two, modified by friction and by the interference of land, gives the actually observed aggregate tide, and it is clear that about new and full moon we must have spring tides and at quarter moons neap tides, and that (the sum of the lunar and solar tide-generating forces being about three times their difference)
So far we have supposed the luminaries to move on the equator; now let us consider the case where the moon is not on the equator. It is clear in this case that at any place the moon's zenith distance at the upper transit is different from her nadir distance at the lower transit. But the tide-generating force is greater the smaller the zenith or nadir distance, and therefore the forces are different at successive transits. This was not the case when the moon was deemed to move on the equator. Thus there is a tendency for two successive lunar tides to be of unequal heights, and the resulting inequality of height is called a "diurnal tide." This tendency vanishes when the moon is on the equator; and, as this occurs each fortnight, the lunar diurnal tide is evanescent once a fortnight. Similarly in summer and winter the successive solar tides are generally of unequal height, whilst in spring and autumn this difference is inconspicuous.
One of the most remarkable conclusions of Laplace's theory of the tides, on a globe covered with ocean to a uniform depth, is that the diurnal tide is everywhere non-existent. But this hypothesis differs much from the reality, and in fact at some ports the diurnal tide is so large that during two portions of each lunation there is only one great high water and one great low water in each twenty-four hours, whilst in other parts of the lunation the usual semi-diurnal tide is observed.
§4. Historical Sketch.
In 1687 Newton laid the foundation for all that has since been added to the theory of the tides when he brought his grand generalization of universal gravitation Kepler, to bear on the subject.Kepler had indeed at an earlier date recognized the tendency of the water of the ocean to move towards the centres of the sun and moon, but he was unable to submit his theory to calculation. Galileo expresses his regret that so acute a man as Kepler should have produced a theory which appeared to him to reintroduce the occult qualities of the ancient philosophers. His own explanation referred the phenomenon to the rotation and orbital motion of the earth, and he considered that it afforded a principal proof of the Copernican system.
In the 19th corollary of the 66th proposition of book i. of the Principia, Newton introduces the conception of a canal circling the earth, and he considers the influence of a satellite on the water in the canal. He remarks that the movement of each molecule of fluid must be accelerated in the conjunction and opposition of the satellite with the molecule, and retarded in the quadratures, so that the fluid must undergo a tidal oscillation. It is, however, in propositions 26 and 27 of book iii. that he first determines the tidal force due to the sun and moon. The sea is here supposed to cover the whole earth, and to assume at each instant a figure of equilibrium, and the tide-generating bodies are supposed to move in the equator. Considering only the action of the sun, he assumes that the figure is an ellipsoid of revolution with its major axis directed towards the sun, and he determines the ellipticity of such an ellipsoid. High solar tide then occurs at noon and midnight, and low tide at sunrise and sunset. The action of the moon produces a similar ellipsoid, but of greater ellipticity. The superposition of these ellipsoids gives the principal variations of tide. He then proceeds to consider the influence of latitude on the height of tide, and to discuss other peculiarities of the phenomenon. Observation shows, however, that spring tides occur a day and a half after syzygies, and Newton falsely attributed this to the fact that the oscillations would last for some time if the attractions of the two bodies were to cease.
The Newtonian hypothesis, although it fails in the form which he gave to it, may still be made to represent the tides, if the lunar and solar ellipsoids have their major axes always directed towards a fictitious moon and sun, which are respectively at constant distances from the true bodies; these distances are such that the syzygies of the fictitious planets occur about a day or a day and a half later than the true syzygies. In fact, the actual tides may be supposed to be generated directly by the action of the real sun and moon, and the wave may be imagined to take a day and a half to arrive at the port of observation. This period has accordingly been called "the age of the Age of tide." In what precedes the planets have been supposed to tide, move in the equator; but the theory of the two ellipsoids cannot be reconciled with the truth when they move in orbits inclined to the equator. At equatorial ports the theory of the ellipsoids would at spring tides give morning and evening high waters of nearly equal height, what ever the declinations of the bodies. But at a port in any other latitude these high waters would be of very different heights, and at Brest, for example, when the declinations of the bodies are equal to the obliquity of the elliptic, the evening tide would be eight times as great as the morning tide. Now observation shows that at this port the two tides are nearly equal to one another, and that their greatest difference is not a thirtieth of their sum.
Newton here also offered an erroneous explanation of the phenomenon. In fact, we shall see that by Laplace's dynamical theory the diurnal tide is evanescent when the ocean is of uniform depth over the earth. At many non-European ports, however, the diurnal tide is very important, and thus as an actual means of prediction the dynamical theory, where the ocean is treated as of uniform depth, may be hardly better than the equilibrium theory.
In 1738 the Academy of Sciences of Paris offered, as a subject for a prize, the theory of the tides. The authors of four essays received prizes, viz., Daniel Bernoulli, Euler, Maclaurin, and Cavalleri. The first three adopted not only the theory of gravitation but also Newton's method of the superposition of the two ellipsoids. Bernoulli's essay contained an extended development of the conception of the two ellipsoids, and, under the name of the equilibrium theory, it is commonly associated with his name. Laplace gives an account and critique of the essays of Bernoulli and Euler in the Mécanique Céleste. The essay of Maclaurin presented little that was new in tidal theory, but is notable as containing those theorems concerning the attraction of ellipsoids which we now know by his name. In 1746 D'Alembert wrote a paper in which he treated the tides of the atmosphere; but this work, like Maclaurin's, is chiefly remarkable for the importance of collateral points.
xxxxment have disappeared through friction is coperiodic with of forced the forces acting on the system. Hence, if the sea is solicited by a periodic force expressed as a coefficient multiplied by the cosine of an angle which increases proportionately with the time, there results a partial tide, also expressed by the cosine of an angle which increases at the same rate; but the phase of the angle and the coefficient of the cosine in the expression for the height may be very different from those occurring in the corresponding term of the equilibrium theory. The coefficients and the constants or epochs of the angles in the expressions for the tide are only derivable from observation. The action of the sun and moon is ex pressible in a converging series of similar cosines; whence there arise as many partial tides, which by the principle of superposition may be added together to give the total tide at any port. In order to unite the several constants of the partial tides Laplace considers each tide as being produced by a fictitious satellite moving uniformly on the equator. Sir W. Thomson and others have followed Laplace in this conception; but in the present article we shall not do so. The difference of treatment is in reality only a matter of phraseology, and the proper motion of each one of Laplace's astres fictifs is at once derivable from the argument (or angle under the sign of cosine), which we shall here associate with the partial tides.The theory of the tidal movements of an ocean was therefore, as Laplace remarks, almost untouched when in 1774 he first undertook the subject. In the Mécanique Céleste he gives an interesting account of the manner in which he was led to attack the problem. We shall give below the investigation of the tides of an ocean covering the whole earth; the theory is substantially Laplace's, although presented in a somewhat different form. This theory, although very wide, is far from representing the tides of our ports. Observation shows, in fact, that the irregular distribution of land and water and the variable depth of the ocean produce an irregularity in the oscillations of the sea of such complexity that the rigorous solution of the problem is altogether beyond the power of analysis. Laplace, however, rested his discussion of tidal observation on this principle-The state of oscillation of a system of bodies in which the primitive conditions of
Subsequently to Laplace the most important workers in this field were Sir John Lubbock (senior), Whewell, and Airy. The Work of Lubbock and Whewell (see §34 below) is chiefly remarkable for the coordination and analysis of enormous masses of data at various ports, and the construction of trustworthy tide-tables and of cotidal maps. Airy contributed an important review of the whole tidal theory. He also studied profoundly the theory of waves in canals, and explained the effects of frictional resistances on the progress of tidal and other waves. Of other authors whose work is of great importance we shall speak below. Amongst all the grand work which has been bestowed on this difficult subject, Newton, notwithstanding his errors, stands out first, and next to him we must rank Laplace. However original any future contribution to the science of the tides may be, it would seem as though it must perforce be based on the work of these two.
A complete list of works bearing on the theory of the tides, from the time of Newton down to 1881, is contained in vol. ii. of the Bibliographie de l'Astronomie by Houzeau and Lancaster (Brussels, 1882). This list does not con tain papers on the tides of particular ports, and we are not aware of the existence of any catalogue of works on practical observation, reduction of observations, prediction, and tidal instruments. References are, however, given below to several works on these points.
- Société de Physique, September 1881, or Proc. Roy. Soc. of Edinburgh, 1881-82, p. 396.
- Founded on Airy's "Tides and Waves," in Ency. Metrop.
- See a series of papers bearing on this kind of wave by Sir W. Thomson, in Phil. Mag., 1886-87.
- Airy, "Tides and Waves."
- Airy, op. cit., §§ 572-573.
- Founded on Laplace, Mécanique Céleste, bk. xiii. chap. i.