Sir William Thomson (Lord Kelvin) (1824-1907), The Tides

Vol. 30, pp. 274-285 of The Harvard Classics

We to-day know that there is a direct relation between the moon and tides. When Julius Cæsar went to conquer Britain his trans­ports were wrecked because he did not know the tides on the English coast; a knowledge of which might have changed the whole course of history.

(Kelvin delivers lecture on "Tides," Aug. 25, 1882.)

[Evening Lecture to the British Association at the Southampton Meeting, Friday, August 25th, 1882]

THE SUBJECT on which I have to speak this evening is the tides, and at the outset I feel in a curiously difficult position. If I were asked to tell what I mean by the Tides I should feel it exceedingly difficult to answer the question. The tides have something to do with motion of the sea. Rise and fall of the sea is sometimes called a tide; but I see, in the Admiralty Chart of the Firth of Clyde, the whole space between Ailsa Craig and the Ayrshire coast marked “very little tide here.” Now, we find there a good ten feet rise and fall, and yet we are authoritatively told there is very little tide. The truth is, the word “tide” as used by sailors at sea means horizontal motion of the water; but when used by landsmen or sailors in port, it means vertical motion of the water. I hope my friend Sir Frederick Evans will allow me to say that we must take the designation in the chart, to which I have referred, as limited to the instruction of sailors navigating that part of the sea, and to say that there is a very considerable landsman’s tide there—a rise and fall of the surface of the water relatively to the land—though there is exceedingly little current.

  One of the most interesting points of tidal theory is the determination of the currents by which the rise and fall is produced, and so far the sailor’s idea of what is most noteworthy as to tidal motion is correct: because before there can be a rise and fall of the water anywhere it must come from some other place, and the water cannot pass from place to place without moving horizontally, or nearly horizontally, through a great distance. Thus the primary phenomenon of the tides is after all the tidal current; and it is the tidal currents that are referred to on charts where we have arrow-heads marked with the statement that we have “very little tide here,” or that we have “strong tides” there.

  One instance of great interest is near Portland. We hear of the “race of Portland” which is produced by an exceedingly strong tidal current; but in Portland harbour there is exceedingly little rise and fall, and that little is much confused, as if the water did not know which way it was going to move. Sometimes the water rises, sinks, seems to think a little while about it, and then rises again. The rise of the tide at Portland is interesting to the inhabitants of Southampton in this, that whereas here, at Southampton, there is a double high water, there, at Portland, there is a double low water. The double high water seems to extend across the channel. At Havre, and on the bar off the entrance to Harve, there is a double high water very useful to navigation; but Southampton I believe is pre-eminent above all the ports in the British Island with respect to this convenience. There is here (at Southampton) a good three hours of high water;—a little dip after the first high water, and then the water rises again a very little more for an hour and a half or two hours, before it begins to fall to low water.

  I shall endeavour to refer to this subject again. It is not merely the Isle of Wight that gives rise to the phenomenon. The influence extends to the east as far as Christchurch, and is reversed at Portland, and we have the double or the prolonged high water also over at Havre; therefore, it is clearly not, as it has been supposed to be, due to the Isle of Wight.

  But now I must come back to the question, What are the “Tides”? Is a “tidal wave” a tide? What is called in the newspapers a “tidal wave” rises sometimes in a few minutes, does great destruction, and goes down again, somewhat less rapidly. There are frequent instances in all parts of the world of the occurrence of that phenomenon. Such motions of the water, however, are not tides; they are usually caused by earthquakes. But we are apt to call any not very short-time and fall of the water a tide, as when standing on the coast of a slanting shore where there are long ocean waves, we see the gradual sinkings and risings produced by them, and say that it is a wave we see, not a tide, till one comes which is exceptionally slow, and then we say “that is liker a tide than a wave.” The fact is, there is something perfectly continuous in the species of motion called wave, from the smallest ripple in a musical glass, whose period may be a thousandth of a second, to a “lop of water” in the Solent, whose period is one or two seconds, and thence on to the great ocean wave with a period of from fifteen to twenty seconds, where end the phenomenon which we commonly call waves (FIG. 122), and not tides. But any rise and fall which is manifestly of longer period, or slower in its rise from lowest to highest, than a wind wave, we are apt to call a tide; and some of the phenomena that are analysed for, and worked out in this very tidal analysis that I am going to explain, are in point of fact more properly wind waves than true tides.

Fig. 122–Wave forms

  Leaving these complicated questions, however, I will make a short cut, and assuming the cause without proving it, define the thing by the cause. I shall therefore define tides thus: Tides are motions of water on the earth, due to the attractions of the sun and of the moon. I cannot say tides are motions due to the actions of the sun and of the moon; for so I would include, under the designation of tide, every ripple that stirs a puddle or millpond, and waves in the Solent or in the English Channel and the long Atlantic wind waves, and the great swell of the ocean from one hemisphere to the other and back again (under the name which I find in the harmonic reduction of tidal observations), proved to take place once a year, and which I can only explain as the result of the sun’s heat.

  But while the action of the sun’s heat by means of the wind produced ripples and waves of every size, it also produces a heaping-up of the water (FIG. 123). Suppose we have wind blowing across one side of a sheet of water, the wind ruffles the surface, the waves break if the wind is strong, and the result is a strong tangential force exerted by the wind on the surface water. If a ship is sailing over the water there is strong tangential force; thus the water is found going fast to leeward for a long distance astern of a great ship sailing with a side wind: and, just as the sails of a ship standing high above the sea give a large area for the wind to act upon, every wave standing up gives a surface, and we have horizontal tangential force over the whole surface of a troubled sea. The result is that water is dragged along the surface from one side of the ocean to the other—from one side of the Atlantic to the other—and is heaped up on the side towards which the wind is blowing. To understand the dynamics of this phenomenon, think of a long straight canal with the wind blowing lengthwise along it. In virtue of the tangential force exerted on the surface of the water by the wind, and which increases with the speed of the wind, the water will become heaped up at one end of the canal, as shown in the diagram (FIG. 123), while the surface water throughout the whole length will be observed moving in the direction of the wind—say in the direction of the two arrows near to the surface of the water above and below it. But to re-establish the disturbed hydrostatic equilibrium, the water so heaped up will tend to flow back to the end from which it has been displaced, and as the wind prevents this taking place by a surface current, there will be set up a return current along the bottom of the canal, in a direction opposite to that of the wind, as indicated by the lowermost arrow in the diagram (FIG. 123). The return current in the ocean, however, is not always an under-current, but may sometimes be a lateral current. Thus a gale of wind blowing over ten degrees of latitude will cause a drag of water at the surface, but the return may be not an under current but a current on one side or the other of the area affected by the wind. Suppose, for instance, in the Mediterranean there is a strong east wind blowing along the African coast, the result will be a current from east to west along that coast, and return current along the northern coasts of the Mediterranean.

Fig. 123–Showing the heaping-up of water produced by wind

  The rise and fall of the water due to these motions are almost inextricably mixed-up with the true tidal rise and fall.

  There is another rise and fall, also connected with the heating effect of the sun, that I do not call a true tide, and that is a rise and fall due to change of atmospheric pressure. When the barometer is high over a large area of ocean, then, there and in neighboring places, the tendency to hydrostatic equilibrium causes the surface of the water to be lower, where it is pushed down by the greater weight of air, and to be higher where there is less weight over it. It does not follow that in every case it is lower, because there may not be time to produce the effect, but there is this tendency. It is very well known that two or three days of low barometer make higher tides on our coast. In Scotland and England and Ireland, two or three days of low barometer generally produce all round the shore higher water than when the barometer is high; and this effect is chiefly noticed at the time of tidal high water, because people take less notice of low water—as at Portland where they think nothing of the double low water. Hence we hear continually of very high tides—very high water noticed at the time of high tides—when the barometer is low. We have not always, however, in this effect of barometric pressure really great tidal rise and fall. On the contrary we have the curious phenomenon that sometimes when the barometer is very low, and there are gales in the neighbourhood, there is very little rise and fall, as the water is kept heaped up and does not sink by anything like its usual amount from the extra high level that it has at high water. But I fear I have got into questions which are leading me away from my subject, and as I cannot get through them I must just turn back.

  Now think of the definition which I gave of the “tides,” and think of the sun alone. The action of the sun cannot be defined as the cause of the solar tides. Solar tides are due to action of the sun, but all risings and fallings of the water due to the action of the sun are not tides. We want the quantification of the predicate here very badly. We have a true tide depending on the sun, the mean solar diurnal tide, having for its period twenty-four solar hours, which is inextricably mixed up with those meteorological tides that I have just been speaking of—tides depending on the sun’s heat, and on the variation of the direction of the wind, and on the variation of barometric pressure according to the time of day. The consequence is that in tidal analysis, when we come to the solar tides, we cannot know how much of the analysed result is due to attraction, and how much to heating effect directly or indirectly, whether on water, or on air, or on water as affected by air. As to the lunar tides we are quite sure of them;—they are gravitational, and nothing but gravitational; but I hope to speak later of the supposed relation of the moon to the weather, and the relation that has to the tides.

  I have defined the tides as motions of water on the earth due to the attractions of the sun and of the moon. How are we to find out whether an observed motion of the water is a tide or is not a tide as thus defined? Only by the combination of theory and observation: if these afford sufficient reason for believing that the motion is due to attraction of the sun or of the moon, or of both, then we must call it a tide.

  It is curious to look back on the knowledge of the tides possessed in ancient times, and to find as early as two hundred years before the Christian era a very clear account given of the tides at Cadiz. But the Romans generally, knowing only the Mediterranean, had not much clear knowledge of the tides. At a much later time than that, we hear from the ancient Greek writers and explorers—Posidonius, Strabo, and others—that in certain remote parts of the world, in Thule, in Britain, in Gaul, and on the distant coasts of Spain, there were motions of the sea—a rising and falling of the water—which depended in some way on the moon. Julius Cæsar came to know something about it; but it is certain the Roman Admiralty did not supply Julius Cæsar’s captains with tide tables when he sailed from the Mediterranean with his expeditionary force, destined to put down anarchy in Britain. He says, referring to the fourth day after his first landing in Britain—”That night it happened to be full moon, which time is accustomed to give the greatest risings of water in the ocean, though our people did not know it.” It has been supposed however that some of his people did know it—some of his quartermasters had been in England before and did know—but that the discipline in the Roman navy was so good that they had no right to obtrude their knowledge; and so, although a storm was raging at the time, he was not told that the water would rise in the night higher than usual, and nothing was done to make his transports secure higher up on the shore while he was fighting the Britons. After the accident Cæsar was no doubt told—”Oh, we knew that before, but it might have been ill taken if we had said so.”

  Strabo says—”Soon after moonrise the sea begins to swell up and flow over the earth till the moon reaches mid heaven. As she descends thence the sea recedes till about moonset, when the water is lowest. It then rises again as the moon, below the horizon, sinks to mid heaven under the earth.” It is interesting here to find the tides described simply with reference to the moon. But there is something more in this ancient account of Strabo; he says, quoting Posidonius—”This is the daily circuit of the sea. Moreover, there is a regular monthly course, according to which the greatest rise and fall takes place about new moon, then diminishing rise and fall till half moon, and again increasing till full moon.” And lastly he refers to a hearsay report of the Gaditani (Cadizians) regarding an annual period in the amount of the daily rise and fall of the sea, which seems to be not altogether right, and is confessedly in part conjectural. He gave no theory, of course, and he avoided the complication of referring to the sun. But the mere mention of an annual period is interesting in the history of tidal theory, as suggesting that the rises and falls are due not to the moon alone but to the sun also. The account given by Posidonius is fairly descriptive of what occurs at the present day at Cadiz. Exactly the opposite would be true at many places; but at Cadiz the time of high water at new and full moon is nearly twelve o’clock. Still, I say we have only definition to keep us clear of ambiguities and errors; and yet, to say that those motions of the sea which we call tides depend on the moon, was considered, even by Galileo, to be a lamentable piece of mysticism which he read with regret in the writings of so renowned an author as Kepler.

  It is indeed impossible to avoid theorising. The first who gave a theory was Newton; and I shall now attempt to speak of it sufficiently to allow us to have it as a foundation for estimating the forces with which we are concerned, in dealing with some of the very perplexing questions which tidal phenomena present.

  We are to imagine the moon as attracting the earth, subject to the forces that the different bodies exert upon each other. We are not to take Hegel’s theory—that the Earth and the Planets do not move like stones, but move along like blessed gods, each an independent being. If Hegel had any grain of philosophy in his ideas of the solar system, Newton is all wrong in his theory of the tides. Newton considered the attraction of the sun upon the earth and the moon, of the earth upon the moon, and the mutual attractions of different parts of the earth; and left it for Cavendish to complete the discovery of gravitation, by exhibiting the mutual attraction of two pieces of lead in his balance. Tidal theory is one strong link in the grand philosophic chain of the Newtonian theory of gravitation. In explaining the tide-generating force we are brought face to face with some of the subtleties, and with some of the mere elements, of physical astronomy. I will not enter into details, as it would be useless for those who already understand the tidal theory, and unintelligible to those who do not.

  I may just say that the moon attracts a piece of matter, for example a pound-weight, here on the earth, with a force which we compare with the earth’s attraction thus. Her mass is 1/80 of the earth’s, and she is sixty times as far away from the earth’s centre as we are here. Newton’s theory of gravitation shows, that when you get outside the mass of the earth the resultant attraction of the earth on the pound weight, is the same as if the whole mass of the earth were collected at the centre, and that it varies inversely as the square of the distance from the centre. The same law is inferred regarding the moon’s attraction from the general theory. The moon’s attraction on this pound weight is therefore 1/80/60x60, or 1/288,000 of the attraction of the earth on the same mass. But that is not the tide-generating force. The moon attracts any mass at the nearest parts of the earth’s surface with greater force than an equal mass near the centre; and attracts a mass belonging to the remoter parts with less force. Imagine a point where the moon is overhead, and imagine another point on the surface of the earth at the other end of a diameter passing through the first point and the centre of the earth (illustrated by B and A of FIG. 124, p. 284). The moon attracts the nearest point (B) with a force which is greater than that with which it attracts the farther point (A) in the ratio of the square of 59 to the square of 61. Hence the moon’s attraction on equal masses at the nearest and farthest points differs by one fifteenth part of her attraction on an equal mass at the earth’s centre, or about a 4,320,000th, or, roughly, a four-millionth, of the earth’s attraction on an equal mass at its surface. Consequently the water tends to protrude towards the moon and from the moon. If the moon and earth were held together by a rigid bar the water would be drawn to the side nearest to the moon—drawn to a prodigious height of several hundred feet. But the earth and the moon are not so connected. We may imagine the earth as falling towards the moon, and the moon as falling towards the earth, but never coming nearer; the bodies, in reality, revolving round their common centre of gravity. A point nearest to the moon is as it were dragged away from the earth, and thus the result is that apparent gravity differs by about one four-millionth at the points nearest to and farthest from the moon. At the intermediate points of the circle C, D (FIG. 124, p. 284), there is a somewhat complicated action according to which gravitation is increased by about one 17-millionth, and its direction altered by about one 17-millionth, so that a pendulum 17,000 feet long, a plummet rather longer than from the top of Mont Blanc to sea level, would, if showing truly the lunar disturbing force, be deflected through a space of one thousandth of a foot. It seems quite hopeless by a plummet to exhibit the lunar disturbance of gravity. A spring balance to show the alteration of magnitude, and a plummet to show the change of direction are conceivable; but we can scarcely believe that either can ever be produced, with sufficient delicacy and consistency and accuracy to indicate these results.

  A most earnest and persevering effort has been made by Mr. George Darwin and Mr. Horace Darwin to detect variations in gravity due to lunar disturbance, and they have made apparatus which, notwithstanding the prodigious smallness of the effect to be observed, is in point of delicacy and consistency capable of showing it; but when they had got their delicate pendulum—their delicate plummet about the length of an ordinary second’s pendulum—and their delicate multiplying gear to multiply the motion of its lower end by about a million times, and to show the result on a scale by the reflection of a ray of light, they found the little image incessantly moving backward and forward on the scale with no consistency or regularity; and they have come to the conclusion that there are continual local variations of apparent gravity taking place for which we know no rule, and which are considerably greater than the lunar disturbance for which they were seeking. That which they found—continual motions of the surface of the earth, and which was not the primary object of their investigation—is in some respects more interesting than what they sought and did not find. The delicate investigation thus opened up promises a rich harvest of knowledge. These disturbances are connected with earthquakes such as have been observed in a very scientific and accurate manner by Milne, Thomas Gray, and Ewing in Japan, and in Italy by many accurate observers. All such observations agree in showing continual tremor and palpitation of the earth in every part.

  One other phenomenon that I may just refer to now as coming out from tide-gauge observations, is a phenomenon called seiches by Forel, and described by him as having been observed in the lakes of Geneva and Constance. He attributes them to differences of barometric pressure at the ends of the lake, and it is probable that part of the phenomenon is due to such differences. I think it is certain, however, that the whole is not due to such differences. The Portland tide curve and those of many other places, notably the tide curve for Malta, taken about ten years ago by Sir Cooper Key, and observations on the Atlantic coasts and in many other parts of the world, show something of these phenomena; a ripple or roughness on the curve traced by the tide gauge, which, when carefully looked to, indicates a variation not regular but in some such period as twenty or twenty-five minutes. It has been suggested that they are caused by electric action! Whenever the cause of a thing is not known it is immediately put down as electrical!

Fig. 124–Spring Tides

Fig. 125–Spring Tides

  I would like to explain to you the equilibrium theory, and the kinetic theory, of the tides, but I am afraid I must merely say there are such things; and that Laplace in his great work, his Mécanique Céleste, first showed that the equilibrium theory was utterly insufficient to account for the phenomena, and gave the true principles of the dynamic action on which they depend. The resultant effect of the tide-generating force is to cause the water to tend to become protuberant towards the moon and the sun and from them, when they are in the same straight line, and to take a regular spheroidal form, in which the difference between the greatest and the least semi-diameter is about 2 feet for lunar action alone, and 1 foot for the action of the sun alone—that is a tide which amounts to 3 feet when the sun and moon act together (FIGS. 124 and 125), and to 1 foot only when they act at cross purposes (FIGS. 126 and 127), so as to produce opposite effects. These diagrams, FIGS. 124 to 127, illustrate spring and neap tides: the dark shading around the globe, E, representing a water envelope surrounding the earth. There has been much discussion on the origin of the word neap. It seems to be an Anglo-Saxon word meaning scanty. Spring seems to be the same as when we speak of plants springing up. I well remember at the meeting of the British Association at Edinburgh a French member who, meaning spring tides, spoke of the grandes marées du printemps. Now you laugh at this; and yet, though he did not mean it, he was quite right, for the spring tides in the spring time are greater on the whole than those at other times, and we have the greatest spring tides in the spring of the year. But there the analogy ceases, for we have also very high spring tides in autumn. Still the meaning of the two words is the same etymologically. Neap tides are scanty tides, and spring tides are tides which spring up to remarkably great heights.

I would like to explain to you the equilibrium theory, and the kinetic theory, of the tides, but I am afraid I must merely say there are such things; and that Laplace in his great work, his Mécanique Céleste, first showed that the equilibrium theory was utterly insufficient to account for the phenomena, and gave the true principles of the dynamic action on which they depend. The resultant effect of the tide-generating force is to cause the water to tend to become protuberant towards the moon and the sun and from them, when they are in the same straight line, and to take a regular spheroidal form, in which the difference between the greatest and the least semi-diameter is about 2 feet for lunar action alone, and 1 foot for the action of the sun alone—that is a tide which amounts to 3 feet when the sun and moon act together (FIGS. 124 and 125), and to 1 foot only when they act at cross purposes (FIGS. 126 and 127), so as to produce opposite effects. These diagrams, FIGS. 124 to 127, illustrate spring and neap tides: the dark shading around the globe, E, representing a water envelope surrounding the earth. There has been much discussion on the origin of the word neap. It seems to be an Anglo-Saxon word meaning scanty. Spring seems to be the same as when we speak of plants springing up. I well remember at the meeting of the British Association at Edinburgh a French member who, meaning spring tides, spoke of the grandes marées du printemps. Now you laugh at this; and yet, though he did not mean it, he was quite right, for the spring tides in the spring time are greater on the whole than those at other times, and we have the greatest spring tides in the spring of the year. But there the analogy ceases, for we have also very high spring tides in autumn. Still the meaning of the two words is the same etymologically. Neap tides are scanty tides, and spring tides are tides which spring up to remarkably great heights.

Fig. 124

Fig.125

Fig. 126

Fig. 127