The computations necessary for this purpose are no doubt simple, so far as their principle is concerned; but they are exceedingly tedious, and any process must be welcomed which affords mitigation of a task so laborious. The entire theory of the tides owes much to Sir William Thomson in the methods of observation and in the methods of reduction. He has now completed the practical parts of the subject by inventing and constructing the famous tide-predicting engine.
The principle of this engine is comparatively simple. There is a chain which at one end is fixed, and at the other end carries the pencil which is pressed against the revolving drum on which the prediction is to be inscribed. Between its two ends the chain passes up and down over pulleys. Each pulley corresponds to one of the “tides,” and there are about a dozen altogether, some of which exercise but little effect. Of course if the centres of the pulleys were all fixed the pen could not move, but the centre of each pulley describes a circle with a radius proportional to the amplitude of the corresponding tide, and in a time proportional to the period of that tide. When these pulleys are all set so as to start at the proper phases, the motion is produced by turning round a handle which makes the drum rotate, and sets all the pulleys in motion. The tide curve is thus rapidly drawn out; and so expeditious is the machine, that the tides of a port for an entire year can be completely worked out in a couple of hours.
While the student or the philosopher who seeks to render any account of the tide on dynamical grounds is greatly embarrassed by the difficulties introduced by friction, we, for our present purpose in the study of the great romance of modern science opened up to us by the theory of the tides, have to welcome friction as the agent which gives to the tides their significance from our point of view.
There is the greatest difference between the height of the rise and fall of the tide at different localities. Out in mid-ocean, for instance, an island like St. Helena is washed by a tide only about three feet in range; an enclosed sea like the Caspian is subject to no appreciable tides whatever, while the Mediterranean, notwithstanding its connection with the Atlantic, is still only subject to very inconsiderable tides, varying from one foot to a few feet. The statement that water always finds its own level must be received, like many another proposition in nature, with a considerable degree of qualification. Long ere one tide could have found its way through the Straits of Gibraltar in sufficient volume to have appreciably affected the level of the great inland sea, its effects would have been obliterated by succeeding tides. On the other hand, there are certain localities which expose a funnel-shape opening to the sea; into these the great tidal wave rushes, and as it passes onwards towards the narrow part, the waters become piled up so as to produce tidal phenomena of abnormal proportions. Thus, in our own islands, we have in the Bristol Channel a wide mouth into which a great tide enters, and as it hurries up the Severn it produces the extraordinary phenomenon of the Bore. The Bristol Channel also concentrates the great wave which gives Chepstow and Cardiff a tidal range of thirty-seven or thirty-eight feet at springs, and forces the sea up the river Avon so as to give Bristol a wonderful tide. There is hardly any more interesting spot in our islands for the observation of tides than is found on Clifton Suspension Bridge. From that beautiful structure you look down on a poor and not very attractive stream, which two hours later becomes transformed into a river of ample volume, down which great ships are navigated. But of all places in the world, the most colossal tidal phenomena are those in the Bay of Fundy. Here the Atlantic passes into a long channel whose sides gradually converge. When the great pulse of the tide rushes up this channel, it is gradually accumulated into a mighty volume at the upper end, the ebb and flow of which at spring tides extends through the astonishing range of not less than fifty feet.
These discrepancies between the tides at different places are chiefly due to the local formations of the coasts and the sea-beds. Indeed, it seems that if the whole earth were covered with an uniform and deep ocean of water, the tides would be excessively feeble. On no other supposition can we reasonably account for the fact that our barometric records fail to afford us any very distinct evidence as to the existence of tides in the atmosphere. For you will, of course, remember that our atmosphere may be regarded as a deep and vast ocean of air, which embraces the whole earth, extending far above the loftiest summits of the mountains.
It is one of the profoundest of nature's laws that wherever friction takes place, energy has to be consumed. Perhaps I ought rather to say transformed, for of course it is now well known that consumption of energy in the sense of absolute loss is impossible. Thus, when energy is expended in moving a body in opposition to the force of friction, or in agitating a liquid, the energy which disappears in its mechanical form reappears in the form of heat. The agitation of water by paddles moving through it warms the water, and the accession of heat thus acquired measures the energy which has been expended in making the paddles rotate. The motion of a liquid of which the particles move among each other with friction, can only be sustained by the incessant degradation of energy from the mechanical form into the lower form of diffused heat. Thus the very fact that the tides are ebbing and flowing, and that there is consequently incessant friction going on among all the particles of water in the ocean, shows us that there must be some great store of energy constantly available to supply the incessant draughts made upon it by the daily oscillation of the tides. In addition to the mere friction between the particles of water, there are also many other ways in which the tides proclaim to us that there is some great hoard of energy which is continually accessible to their wants. Stand on the bank of an estuary or river up and down which a great tidal current ebbs and flows; you will see the water copiously charged with sediment which the tide is bearing along. Engineers are well aware of the potency of the tide as a vehicle for transporting stupendous quantities of sand or mud. A sand-bank impedes the navigation of a river; the removal of that sand-bank would be a task, perhaps, conceivably possible by the use of steam dredges and other appliances, whereby vast quantities of sand could be raised and transported to another locality where they would be innocuous. It is sometimes possible to effect the desired end by applying the power of the tide. A sea-wall judiciously thrown out will sometimes concentrate the tide into a much narrower channel. Its daily oscillations will be accomplished with greater vehemence, and as the tide rushes furiously backwards and forwards over the obstacle, the incessant action will gradually remove it, and the impediment to navigation may be cleared away. Here we actually see the tides performing a piece of definite and very laborious work, to accomplish which by the more ordinary agents would be a stupendous task.
In some places the tides are actually harnessed so as to accomplish useful work. I have read that underneath old London Bridge there used formerly to be great water-wheels, which were turned by the tide as it rushed up the river, and turned again, though in the opposite way, by the ebbing tide. These wheels were, I believe, employed to pump up water, though it does not seem obvious for what purposes the water would have been suitable. Indeed in the ebb and flow all round our coasts there is a potential source of energy which has hitherto been allowed to run to waste. The tide could be utilized in various ways. Many of you will remember the floating mills on the Rhine. They are vessels like paddle steamers anchored in the rapid current. The flow of the river makes the paddles rotate, and thus the machinery in the interior is worked. Such craft moored in a rapid tide-way could also be made to convey the power of the tides into the mechanism of the mill. Or there is still another method which has been employed, and which will perhaps have a future before it in those approaching times when the coal-cellars of England shall be exhausted. Imagine on the sea-coast a large flat extent which is inundated twice every day by the tide. Let us build a stout wall round this area, and provide it with a sluice-gate. Open the gate as the tide rises, and the great pond will be filled; then at the moment of high water close the sluice, and the pond-full will be impounded. If at low tide the sluice be opened the water will rush tumultuously out. Now suppose that a water-wheel be provided, so that the rapid rush of water from the exit shall fall upon its blades; then a source of power is obviously the result.
At present, however, such a contrivance would naturally find no advocates, for of course the commercial aspect of the question is that which will decide whether the scheme is practicable and economical. The issue indeed can be very simply stated. Suppose that a given quantity of power be required—let us say that of one hundred horse. Then we have to consider the conditions under which a contrivance of the kind we have sketched shall yield a power of this amount. Sir William Thomson, in a very interesting address to the British Association at York in 1881, discussed this question, and I shall here make use of the facts he brought forward on that occasion. He showed that to obtain as much power as could be produced by a steam-engine of one hundred horse power, a very large reservoir would be required. It is doubtful indeed whether there would be many localities on the earth which would be suitable for the purpose. Suppose, however, an estuary could be found which had an area of forty acres; then if a wall were thrown across the mouth so that the tide could be impounded, the total amount of power that could be yielded by a water-wheel worked by the incessant influx and efflux of the tide would be equal to that yielded by the one hundred horse engine, running continuously from one end of the year to the other.
There are many drawbacks to a tide-mill of this description. In the first place, its situation would naturally be far removed from other conveniences necessary for manufacturing purposes. Then too there is the great irregularity in the way in which the power is rendered available. At certain periods during the twenty-four hours the mill would stop running, and the hours when this happened would be constantly changing. The inconvenience from the manufacturer's point of view of a deficiency of power during neap tides might not be compensated by the fact that he had an excessive supply of power at spring-tides. Before tide-mills could be suitable for manufacturing purposes, some means must be found for storing away the energy when it is redundant, and applying it when its presence is required. We should want in fact for great sources of energy some contrivance which shall fulfil the same purpose as the accumulators do in an electrical installation.
Even then, however, the financial consideration remains, as to whether the cost of building the dam and maintaining the tide-mill in good order will not on the whole exceed the original price and the charges for the maintenance of a hundred horse power steam-engine. There cannot be a doubt that in this epoch of the earth's history, so long as the price of coal is only a few shillings a ton, the tide-mill, even though we seem to get its power without current expense, is vastly more expensive than a steam-engine. Indeed, Sir William Thomson remarks, that wherever a suitable tidal basin could be found, it would be nearly as easy to reclaim the land altogether from the sea. And if this were in any locality where manufactures were possible, the commercial value of forty acres of reclaimed land would greatly exceed all the expenses attending the steam-engine. But when the time comes, as come it apparently will, that the price of coal shall have risen to several pounds a ton, the economical aspect of steam as compared with other prime movers will be greatly altered; it will then no doubt be found advantageous to utilize great sources of energy, such as Niagara and the tides, which it is now more prudent to let run to waste.
For my argument, however, it matters little that the tides are not constrained to do much useful work. They are always doing work of some kind, whether that be merely heating the particles of water by friction, or vaguely transporting sand from one part of the ocean to the other. Useful work or useless work are alike for the purpose of my argument. We know that work can never be done unless by the consumption or transformation of energy. For each unit of work that is done—whether by any machine or contrivance, by the muscles of man or any other animal, by the winds, the waves, or the tides, or in any other way whatever—a certain equivalent quantity of energy must have been expended. When, therefore, we see any work being performed, we may always look for the source of energy to which the machine owes its efficiency. In fact, it is the old story illustrated, that perpetual motion is impossible. A mechanical device, however ingenious may be the construction, or however accurate the workmanship, can never possess what is called perpetual motion. It is needless to enter into details of any proposed contrivance of wheels, of pumps, of pulleys; it is sufficient to say that nothing in the shape of mechanism can work without friction, that friction produces heat, that heat is a form of energy, and that to replace the energy consumed in producing the heat there must be some source from which the machine is replenished if its motion is to be continued indefinitely.
Hence, as the tides may be regarded as a machine doing work, we have to ascertain the origin of that energy which they are continually expending. It is at this point that we first begin to feel the difficulties inherent in the theory of tidal evolution. I do not mean difficulties in the sense of doubts, for up to the present I have mentioned no doubtful point. When I come to such I shall give due warning. By difficulties I now mean points which it is not easy to understand without a little dynamical theory; but we must face these difficulties, and endeavour to elucidate them as well as we can.
Let us first see what the sources of energy can possibly be on which the tides are permitted to draw. Our course is simplified by the fact that the energy of which we have to speak is of a mechanical description, that is to say, not involving heat or other more obscure forms of energy. A simple type of energy is that possessed by a clock-weight after the clock has been wound. A store of power is thus laid up which is gradually doled out during the week in small quantities, second by second, to sustain the motion of the pendulum. The energy in this case is due to the fact that the weight is attracted by the earth, and is yielded according as the weight sinks downwards. In the separation between two mutually attracting bodies, a store of energy is thus implied. What we learn from an ordinary clock may be extended to the great bodies of the universe. The moon is a gigantic globe separated from our earth by a distance of 240,000 miles. The attraction between these two bodies always tends to bring them together. No doubt the moon is not falling towards the earth as the descending clock-weight is doing. We may, in fact, consider the moon, so far as our present object is concerned, to be revolving almost in a circle, of which the earth is the centre. If the moon, however, were to be stopped, it would at once commence to rush down towards the earth, whither it would arrive with an awful crash in the course of four or five days. It is fortunately true that the moon does not behave thus; but it has the ability of doing so, and thus the mere separation between the earth and the moon involves the existence of a stupendous quantity of energy, capable under certain conditions of undergoing transformation.
There is also another source of mechanical energy besides that we have just referred to. A rapidly moving body possesses, in virtue of its motion, a store of readily available energy, and it is easy to show that energy of this type is capable of transformation into other types. Think of a cannon-ball rushing through the air at a speed of a thousand feet per second; it is capable of wreaking disaster on anything which it meets, simply because its rapid motion is the vehicle by which the energy of the gunpowder is transferred from the gun to where the blow is to be struck. Had the cannon been directed vertically upwards, then the projectile, leaving the muzzle with the same initial velocity as before, would soar up and up, with gradually abating speed, until at last it reached a turning-point, the elevation of which would depend upon the initial velocity. Poised for a moment at the summit, the cannon-ball may then be likened to the clock-weight, for the entire energy which it possessed by its motion has been transformed into the statical energy of a raised weight. Thus we see these two forms of energy are mutually interchangeable. The raised weight if allowed to fall will acquire velocity, or the rapidly moving weight if directed upwards will acquire altitude.
The quantity of energy which can be conveyed by a rapidly moving body increases greatly with its speed. For instance, if the speed of the body be doubled, the energy will be increased fourfold, or, in general, the energy which a moving body possesses may be said to be proportional to the square of its speed. Here then we have another source of the energy present in our earth-moon system; for the moon is hurrying along in its path with a speed of two-thirds of a mile per second, or about twice or three times the speed of a cannon-shot. Hence the fact that the moon is continuously revolving in a circle shows us that it possesses a store of energy which is nine times as great as that which a cannon-ball as massive as the moon, and fired with the ordinary velocity, would receive from the powder which discharged it.
Thus we see that the moon is endowed with two sources of energy, one of which is due to its separation from the earth, and the other to the speed of its motion. Though these are distinct, they are connected together by a link which it is important for us to comprehend. The speed with which the moon revolves around the earth is connected with the moon's distance from the earth. The moon might, for instance, revolve in a larger circle than that which it actually pursues; but if it did so, the speed of its motion would have to be appropriately lessened. The orbit of the moon might have a much smaller radius than it has at present, provided that the speed was sufficiently increased to compensate for the increased attraction which the earth would exercise at the lessened distance. Indeed, I am here only stating what every one is familiar with under the form of Kepler's Law, that the square of the periodic time is in proportion to the cube of the mean distance. To each distance of the moon therefore belongs an appropriate speed. The energy due to the moon's position and the energy due to its motion are therefore connected together. One of these quantities cannot be altered without the other undergoing change. If the moon's orbit were increased there would be a gain of energy due to the enlarged distance, and a loss of energy due to the diminished speed. These would not, however, exactly compensate. On the whole, we may represent the total energy of the moon as a single quantity, which increases when the distance of the moon from the earth increases, and lessens when the distance from the earth to the moon lessens. For simplicity we may speak of this as moon-energy.
But the most important constituent of the store of energy in the earth-moon system is that contributed by the earth itself. I do not now speak of the energy due to the velocity of the earth in its orbit round the sun. The moon indeed participates in this equally with the earth, but it does not affect those mutual actions between the earth and moon with which we are at present concerned. We are, in fact, discussing the action of that piece of machinery the earth-moon system; and its action is not affected by the circumstance that the entire machine is being bodily transported around the sun in a great annual voyage. This has little more to do with the action of our present argument than has the fact that a man is walking about to do with the motions of the works of the watch in his pocket. We shall, however, have to allude to this subject further on.
The energy of the earth which is significant in the earth-moon theory is due to the earth's rotation upon its axis. We may here again use as an illustration the action of machinery; and the special contrivance that I now refer to is the punching-engine that is used in our ship-building works. In preparing a plate of iron to be riveted to the side of a ship, a number of holes have to be made all round the margin of the plate. These holes must be half an inch or more in diameter, and the plate is sometimes as much as, or more than, half an inch in thickness. The holes are produced in the metal by forcing a steel punch through it; and this is accomplished without even heating the plate so as to soften the iron. It is needless to say that an intense force must be applied to the punch. On the other hand, the distance through which the punch has to be moved is comparatively small. The punch is attached to the end of a powerful lever, the other end of the lever is raised by a cam, so as to depress the punch to do its work. An essential part of the machine is a small but heavy fly-wheel connected by gearing with the cam.
This fly-wheel when rapidly revolving contains within it, in virtue of its motion, a large store of energy which has gradually accumulated during the time that the punch is not actually in action. The energy is no doubt originally supplied from a steam-engine. What we are especially concerned with is the action of the rapidly rotating wheel as a reservoir in which a large store of energy can be conveniently maintained until such time as it is wanted. In the action of punching, when the steel die comes down upon the surface of the plate, a large quantity of energy is suddenly demanded to force the punch against the intense resistance it experiences; the energy for this purpose is drawn from the store in the fly-wheel, which experiences no doubt a check in its velocity, to be regained again from the energy of the engine during the interval which elapses before the punch is called on to make the next hole.
Another illustration of the fly-wheel on a splendid scale is seen in our mighty steel works, where ponderous rails are being manufactured. A white-hot ingot of steel is presented to a pair of powerful rollers, which grip the steel, and send it through at the other side both compressed and elongated. Tremendous power is required to meet the sudden demand on the machine at the critical moment. To obtain this power an engine of stupendous proportions is sometimes attached directly to the rollers, but more frequently an engine of rather less horse-power will be used, the might of this engine being applied to giving rapid rotation to an immense fly-wheel, which may thus be regarded as a reservoir full of energy. The rolling mills then obtain from this store in the fly-wheel whatever energy is necessary for their gigantic task.
These illustrations will suffice to show how a rapidly rotating body may contain energy in virtue of its rotation, just as a cannon-ball contains energy in virtue of its speed of translation, or as a clock-weight has energy in virtue of the fact that it has some distance to fall before it reaches the earth. The rotating body need not necessarily have the shape of a wheel—it may be globular in form; nor need the axes of rotation be fixed in bearings, like those of the fly-wheel; nor of course is there any limit to the dimensions which the rotating body may assume. Our earth is, in fact, a vast rotating body 8000 miles in diameter, and turning round upon its axis once every twenty-three hours and fifty-six minutes. Viewed in this way, the earth is to be regarded as a gigantic fly-wheel containing a quantity of energy great in correspondence with the earth's mass. The amount of energy which can be stored by rotation also depends upon the square of the velocity with which the body turns round; thus if our earth turned round in half the time which it does at present, that is, if the day was twelve hours instead of twenty-four hours, the energy contained in virtue of that rotation would be four times its present amount.
Reverting now to the earth-moon system, the energy which that system contains consists essentially of two parts—the moon-energy, whose composite character I have already explained, and the earth-energy, which has its origin solely in the rotation of the earth on its axis. It is necessary to observe that these are essentially distinct—there is no necessary relation between the speed of the earth's rotation and the distance of the moon, such as there is between the distance of the moon and the speed with which it revolves in its orbit.
For completeness, it ought to be added that there is also some energy due to the moon's rotation on its axis, but this is very small for two reasons: first, because the moon is small compared with the earth, and second, because the angular velocity of the moon is also very small compared with that of the earth. We may therefore dismiss as insignificant the contributions from this source of energy to the sum total.
I have frequently used illustrations derived from machinery, but I want now to emphasize the profound distinction that exists between the rotation of the earth and the rotation of a fly-wheel in a machine shop. They are both, no doubt, energy-holders, but it must be borne in mind, that as the fly-wheel doles out its energy to supply the wants of the machines with which it is connected, a restitution of its store is continually going on by the action of the engine, so that on the whole the speed of the fly-wheel does not slacken. The earth, however, must be likened to a fly-wheel which has been disconnected with the engine. If, therefore, the earth have to supply certain demands on its accumulation of energy, it can only do so by a diminution of its hoard, and this involves a sacrifice of some of its speed.
In the earth-moon system there is no engine at hand to restore the losses of energy which are inevitable when work has to be done. But we have seen that work is done; we have shown, in fact, that the tides are at present doing work, and have been doing work for as long a period in the past as our imagination can extend to. The energy which this work has necessitated can only have been drawn from the existing store in the system; that energy consists of two parts—the moon-energy and the earth's rotation energy. The problem therefore for us to consider is, which of these two banks the tides have drawn on to meet their constant expenditure. This is not a question that can be decided offhand; in fact, if we attempt to decide it in an offhand manner we shall certainly go wrong. It seems so very plausible to say that as the moon causes the tides, therefore the energy which these tides expend should be contributed by the moon. But this is not the case. It actually happens that though the moon does cause the tides, yet when those tides consume energy they draw it not from the distant moon, but from the vast supply which they find ready to their hand, stored up in the rotation of the earth.
The demonstration of this is not a very simple matter; in fact, it is so far from being simple that many philosophers, including some eminent ones too, while admitting that of course the tides must have drawn their energy from one or other or both of these two sources, yet found themselves unable to assign how the demand was distributed between the two conceivable sources of supply.
We are indebted to Professor Purser of Belfast for having indicated the true dynamical principle on which the problem depends. It involves reasoning based simply on the laws of motion and on elementary mathematics, but not in the least involving questions of astronomical observation. It would be impossible for me in a lecture like this to give any explanation of the mathematical principles referred to. I shall, however, endeavour by some illustrations to set before you what this profound principle really is. Were I to give it the old name I should call it the law of the conservation of areas; the more modern writers, however, speak of it as the conservation of moment of momentum, an expression which exhibits the nature of the principle in a more definite manner.
I do not see how to give any very accurate illustration of what this law means, but I must make the attempt, and if you think the illustration beneath the dignity of the subject, I can only plead the difficulty of mathematics as an excuse. Let us suppose that a ball-room is fairly filled with dancers, or those willing to dance, and that a merry waltz is being played; the couples have formed, and the floor is occupied with pairs who are whirling round and round in that delightful amusement. Some couples drop out for a while and others strike in; the fewer couples there are the wider is the range around which they can waltz, the more numerous the couples the less individual range will they possess. I want you to realize that in the progress of the dance there is a certain total quantity of spin at any moment in progress; this spin is partly made up of the rotation by which each dancer revolves round his partner, and partly of the circular orbit about the room which each couple endeavours to describe. If there are too many couples on the floor for the general enjoyment of the dance, then both the orbit and the angular velocity of each couple will be restricted by the interference with their neighbours. We may, however, assert that so long as the dance is in full swing the total quantity of spin, partly rotational and partly orbital, will remain constant. When there are but few couples the unimpeded rotation and the large orbits will produce as much spin as when there is a much larger number of couples, for in the latter case the diminished freedom will lessen the quantity of spin produced by each individual pair. It will sometimes happen too that collision will take place, but the slight diversions thus arising only increase the general merriment, so that the total quantity of spin may be sustained, even though one or two couples are placed temporarily hors de combat. I have invoked a ball-room for the purpose of bringing out what we may call the law of the conservation of spin. No matter how much the individual performers may change, or no matter what vicissitudes arise from their collision and other mutual actions, yet the total quantity of spin remains unchanged.