The Rhythmic Movements Of The Ocean, Tides, Swell, And Waves

( Originally Published 1915 )

THE manifestations of the life of the terrestrial globe, its general movements, the perturbations it is subjected to, the spasms of its crust, and the manifestation of electricity and magnetism which traverse it, as well as all that communicates this incessant restlessness to the Earth, have their origin in the Sun.

But our globe is not composed entirely of its lithosphere; the hydrosphere which covers more than three-fourths of its surface has an importance of which we have already seen something, for example, in connection with the distribution of terrestrial magnetism. We shall now study the movements of the hydrosphere, and here again we shall find evidence of solar influence, either the direct action of the attraction of its mass, or the indirect action of the heating of the molecules of the fluid substances water and air, enveloping the Earth's crust, and from which results the general movements that are the origin of the circulation of the oceans and the circulation of the atmosphere.

It is hardly necessary to recall the facts concerning the importance of the sea in the general economy of the globe. In the first place it occupies more than two-thirds of the surface; out of the 510 millions of square kilometres [194 millions sq. miles] of the terrestrial crust 365 millions [138 millions sq. miles] are covered by water. There is thus far more water than land, and to the oceans, like bodies elected by universal suffrage, must accrue the rights of the majority. The great atmospheric conditions become established, not above the smaller part of the Earth's surface exhibiting the innumerable accidents of the geographical relief, but above the vast uniform oceanic surface the molecules of which freely obey the laws of fluid mechanics.

The total volume of the water of the oceans is about 1300 million cubic kilometres [309 million cubic miles], while that of the emergent dry land is only loo million [24 million cubic miles]. The mean depth of the seas is about 3550 metres [2.2 miles]. All this mass of water contains a quantity of salts in solution and also, doubtless, metals in a state of extremely fine division. This will be readily understood since, the seas, in the first instance, were formed by the collection of the boiling water-streams which condensed from the Earth's primitive atmosphere and were precipitated on the scarcely solidified crust. In such conditions, the water would have dissolved all that was soluble on the surface of the Earth. Sea-water should thus contain all known sub-stances, at any rate in traces. The mean quantity of salts contained in a kilogram [2 lb. 3 oz. 4 dr.] of sea-water is 35 grams [1.25 oz.] and 75% of this total salinity is composed of sodium chloride, that is to say, common salt. The salt in solution in all the seas would provide enough material to construct the African continent in all its relief. With the gold, of which only a few milligrams [a few hundredths of a grain] are contained in a ton of water, a block could be made which, if divided equally among every inhabitant of the Earth, 1,500,000,000 in number, would give to each one an ingot of 40,000 kilograms [44 tons] of the precious metal, or in other words a fortune of 120 million francs [24 million dollars] !

The salinity of the oceans increases their density; a litre [1.05 quarts] of sea-water weighs 1 kilogram [2 lb. 3 oz. 4 dr.] and 28 grams [432.1 gr.] instead of i kilogram as pure water does. One can therefore swim more easily in it as it possesses a greater buoyancy.

The greatest depth revealed in the course of the Pacific soundings is actually 9750 metres [6 miles]. The continents seem to lie on a kind of base, known as the continental plateau, the mean distance of which below the surface is 200 metres [650 ft.]. Beyond the immediate neighbourhood of the continents, the depth rapidly increases, and this applies equally to the Atlantic, the Pacific, and the Indian oceans. In the case of the Mediterranean, the Straits of Gibraltar, dominated by the Rock, impose special temperature conditions, but in all other cases the temperature falls in proportion to the depth below the surface, and when depths of 6000, 7000 and 8000 metres [3.75, 4.35, 5 miles] are reached it is found that the water there has a uniform temperature in the neighbourhood of zero [0°C. = 32° F.]. Here we find ourselves in the presence of one of the paradoxes of terrestrial physics. If we bored into the Earth's crust to a depth of 8000 metres [5 miles] we should find a temperature of about 240°–250° C. [400°–420° F.] at the bottom of the shaft so made, while at the same depth under the seas the result is o° C. [320 F.]! The difficulty is increased by the fact that the crust, which is presumably of less thickness under the oceans, consequently offers less resistance to the trans-mission of the internal heat. The explanation is doubtless to be sought for in the stream of cold water coming from the polar regions, which on account of its greater density falls to the bottom of the great oceanic hollows. Then when the water becomes warmed to above zero [32° F.], it rises by a process of convection, thus producing a vertical oceanic circulation ; the water so coming up is replaced by more cold water from the polar regions.

The movements of these waters constitute one of the most imposing manifestations of the "life" of the globe; by passing a few days on the Brittany coast we can not only admire the magnificent phenomena of the tides, but are also enabled to conceive the immediately apparent laws governing them.

We see, at a certain time, the level of the sea rise in a continuous manner, constituting what is called the rising tide; at the same time the water of the open sea advances towards the land, and this current is known as the flood-tide or the flow. Gradually, the rise of level stops and the flow ceases, the water finally remains stationary at its highest point. It is now the time of high tide.

Subsequently the current recommences in the opposite direction, the water flowing from the land towards the sea; this is the ebb. The level of the sea gradually falls, and that part of the land covered by the rising tide reappears. The falling tide becomes more rapid up to a certain limit, and then its rate decreases and finally ceases altogether, the water having reached its lowest level. This is the epoch of low tide. Soon the process of rise begins again and passes through all its former phases, constituting a second high tide, followed by a second low tide some hours after.

If the phenomenon be watched for several days it may be shown that there are, broadly speaking, two high tides and two low tides daily, but the interval of time between them is not an exact subdivision of a day. Thus, if a high tide be noted at eight o'clock on the morning of a certain day, the high tide of the next day will not occur at eight o'clock but at ten minutes to nine. The interval separating the two high tides which takes place during the same day is not 12 hours but 12 hours and 25 minutes, the difference being half the above. The diurnal period of the tide is thus 24 hours 5o minutes. Now, this is precisely the value of the interval of time separating two successive passages of the Moon over the meridian of the place. We hence see that the Moon is a dominating factor in tide-production.

It will further be observed that at any one place on the coast the water at the moment of highest tide is never at quite the same level on any two following days. For several days the tides increase in height from one day to another; this is the period of spring-tides. Then follows a period when the level of the high tides is less each day; this is the time of neap-tides. Observation shows that the periodicity of spring-tides and neap-tides is the same as that of the phases of the Moon, and these latter depend on the relative positions of the Moon and the Sun with regard to the Earth. Consequently, although the Moon is the principal factor which governs the tide period, the Sun also has an effect which modifies the magnitude of the phenomenon.

There is another observed fact which is note-worthy, viz., that the height of the tide may have very different values on the same day at two points of the Earth which are close to one another and which are therefore at the same distance from the attracting bodies. For example, if we find a tide at Granville, of 6.11 metres [20 ft.] on a certain day, the height of the tide at the neighbouring port of Cherbourg on the same day will only be 2.82 metres [9.2 ft.]. There is, therefore, a geographical factor which influences the phenomenon and which arises from the coastal con-figuration at the place considered. Finally, it may be proved that high tide at any given place does not take place exactly according to the astronomical attractions arising from the position of the Moon and of the Sun; it takes place some time afterwards and the time-interval of retardation is constant for each place. For any port this is called the establishment of the port. The greatest establishment of the port in France is 12 hours 30 minutes, at Dunkerque ; the smallest is at Lorient, viz., 3 hours 32 minutes. Here again the geographical configuration of the coasts and the irregularity of the sea-bottom have an important influence.

It was Newton who first gave the explanation of the beautiful phenomena of the tides. The Moon, on account of its proximity to the Earth, attracts the molecules of the water in the oceans that are situated on the side of the Earth facing it, to a greater extent than it attracts the centre of the globe and this latter is attracted more than the molecules of water situated on the opposite side of the Earth. We, thus, find at the free surfaces of the seas two liquid protuberances, the summits of which are situated on the line which joins the centre of the Moon to the centre of the Earth. The first is due to the attraction towards the Moon of the fluid mass lying on the side nearest it; the second, on the opposite side arises from the fact that the centre of the globe is more strongly attracted, being nearer the Moon, than the water on the far side which is, so to speak, left behind and hence forms a protuberance (Fig. 27).

When the Sun is in the same direction as the Moon with respect to the Earth, that is to say at the epoch of syzygy, their attractive forces are additive; when the two bodies have their centres on the two sides of a right angle formed at the centre of the Earth, that is to say at the epoch of quadrature, the attractions oppose one another. We have seen, in studying the deviation from the vertical under the influence of the luni-solar attraction, that if we represent the Sun's attraction by 1, that of the Moon is approximately equal to 2. The tide would thus have theoretically for relative amplitudes 2 + I, i. e., 3 at the epochs of spring-tides and 2 -1, i.e., I at the epochs of neap-tides.

This explanation of Newton, based on the equilibrium between the lunar attraction and gravity, gives an account of the phenomenon in its broad outlines and general details, but is found wanting when certain observational facts are taken into consideration. For example, if we apply the theory of static equilibrium to the calculation of the height of the liquid protuberance which represents the tide, we obtain a result of 35 centimetres [13.75 in.]. Now, the most cursory observation shows that the variations of the level of the sea under tidal influence have much greater values than this. In the ports of the English Channel and Brittany the variation is several metres; at Mont Saint-Michel, at the epoch of spring-tides, the difference reaches 14 metres [46 ft.], while in the Straits of Magellan it attains 18 metres [59 ft.], and in the Bay of Fundy, on the coast of Newfoundland, 21 metres [69 ft.].

Furthermore, in certain regions, for example, in Polynesia or in the Gulf of Tonkin, there is only one tide daily instead of two.

Consequently the simple consideration of equilibrium between the astronomical attractive forces and the terrestrial gravity do not suffice to explain the amplitude of the tide; neither do they satisfactorily account for the observed retardations, nor for the differences between two neighbouring places. It will, therefore, be necessary to seek for the complete explanation of the tides, not in the law of fluid equilibrium, but in that of their movements, or in other words not in the study of hydrostatics, but in that of hydrodynamics.

We shall once more find Laplace's genius the starting point of this theory. The illustrious mathematician recognised that when the lunisolar attraction influences the water of the oceans, this attraction, which is exercised by two bodies that move with respect to the Earth, should give rise not to a fixed protuberance, a liquid hill so to speak, but to a true undulation which would move over the sea surface in accordance with some more or less complex law, depending on the nature of the relative movements of the attracting bodies, on the angular variations of their positions with regard to the Earth, and on the variations of actual distance from the Earth. Thus, since we are led to consider wave propagation, we must study the matter from the point of view of fluid dynamics, taking account of the resistances of fluids. Two fundamental principles govern this study, first that of the superposition of small movements and secondly, that of periodicity. The enunciation of the first is as follows: Let us start with the assumption that a system of material points is in equilibrium, and that a very small force is applied so as to disturb this equilibrium. Then, a material point will be given a small velocity, so small that the expression of the force depends only on the time and the mean position of the point. In these conditions, if several forces act simultaneously, the laws of mechanics show that at each instant their effects are independent and consequently superimposable. Also, as these momentary effects have no influence upon the forces themselves, it follows that the total effect will be the sum of the partial efforts calculated as if each force acted separately.

This first principle is of extreme importance; it enables us to consider separately the influence of the Moon and the Sun. That is if we evaluate on the one hand the solar tide and on the other hand the lunar tide, we get the resultant tide by adding the two together. All optical and acoustic phenomena are illustrations of this principle; light and sound waves proceed through space without mutual interference, and, at the present time, trains of electric waves rapidly travel above the Earth's surface, propagating themselves in all directions without the one in any way hindering the others.

The second principle is that of the periodicity of movements caused by periodic forces: Every periodic force produces periodic movements in the group of molecules on which it acts. The periods of the force and resulting movement are equal and at a given point their difference of phase is constant. Thus, a body which travels uniformly in the plane of the equator, remaining always at an invariable distance from the Earth, would give rise, by its relative diurnal movement, to perturbing forces having a period of half a day. In any place whatever, the variations of the level of the sea resulting from this action would have, according to the second principle above, exactly the same semi-diurnal period, and the diverse phases of the movements would be displaced relatively to the corresponding phases of the force by a constant interval of time.

The relative movement of the attracting bodies, however, does not take place in so simple a manner as this, and, leaving the other elements out of consideration, the fact of the inclination of their apparent paths to the plane of the terrestrial equator necessitates the addition of two other categories of forces to those of semi-diurnal period. These other forces are diurnal and long period forces respectively.

In the first place, the action of a body turning only around the Earth will be of semi-diurnal period, for the result is the production of two tide wave-summits diametrically opposite one another. The terrestrial diameter which joins these two summits follows the orbital movement of the body. In these circumstances, the two waves travel around the Earth in such a way that, in the course of the twenty-four hours, which is by supposition the period of the body's movement, a point A of the Earth will be twice affected by the tide wave, viz., once by the wave M and once by the wave M' (Fig. 27). The period will be semi-diurnal. This is what would occur if the body was in the plane of the equator and at a constant distance from the Earth.

But the body is not usually in the plane of the terrestrial equator. In the general case it is outside this plane, either above or below it. Consequently the two zones of deformation M and M', through which every point of the Earth passes in its diurnal movement, are unsymmetrical. These two protuberances have therefore unequal effects, and this inequality is represented analytically by the superimposition of a movement of the sea, of diurnal period, on the semi-diurnal wave of which we have previously spoken. We already see that there will be a considerable number of waves; in fact, if we take as unity the solar day, we shall have, on the one hand, to express the Sun's action, a solar day of 24 hours and a semi-solar day, and for the Moon, on the other hand, a lunar day of 24 hours 50 minutes and a semi-lunar day of 12 hours 25 minutes.

The orbits of the two bodies are, furthermore, inclined to the plane of the terrestrial equator; the charges to which the declinations of the Moon and Sun are subjected from one day to the next are accompanied by little variations in the respective durations of the true solar day and the lunar day. This has an effect on the amplitude and period of the diurnal and semi-diurnal undulations. We must next note that the Moon and the Sun are not at constant distances from the Earth, owing to the ellipticity of the orbits of the Moon and Earth. There is a variation to the extent of 1/18 part in the case of the Moon in a month, and one of 1/60 part in the distance of the Sun from the Earth in a year. This is another cause of modification of the amplitude of the solar and lunar undulations, and the result is as if we combined with each of the principal waves, of their average amplitude, two subsidiary waves having as amplitude the semi-difference between the mean and the maximum, and for respective velocities the sum of and the difference between the principal wave and the variation of amplitude.

We have already mentioned that long period waves are superimposed upon these diurnal and semi-diurnal ones, and the period of each of these new waves depends on the duration of the apparent revolution of each of the attracting bodies round the Earth. Consequently, for the Moon there is a fortnightly wave and a monthly wave, and for the Sun a half-yearly wave and a yearly one, but their amplitudes are much less than those of the principal waves.

This is not all; we have seen in studying the Earth's movements that the place of the lunar orbit suffers a recurrent displacement, and that its intersection with the plane of the terrestrial orbit is continually being displaced from west to east, executing a complete turn in 181 years. As this must be taken into account in considering the inclination of the lunar orbit with respect to the terrestrial one, it will readily be understood that two new subsidiary waves are introduced, the one having a period of 18i years and the other a period of half this, but being of very small amplitude. Other causes affecting the tides are the precession of the equinoxes, the Earth's polar displacements, and, in general, all the Earth's movements, but these waves are negligible in amplitude.

The complete understanding of the tides thus necessitates the calculation of the elements of each of these waves separately and the final combination of their periodic actions. Lord Kelvin and Sir George Darwin have carried out this work, using the beautiful method that is called " harmonic analysis," and, in France, the hydrographer Matt has also made a careful study of the matter.

In order to represent each one of these waves separately, we imagine for each one a fictitious astronomical body, supposed to be the only body in the Earth's presence, turning around the latter with a period equal to that of the wave it re-presents, remaining always in the plane of the equator and at a constant distance from the terrestrial globe. This fictitious body, the mass and distance of which may be calculated from astronomical data, therefore always produces a constant, unique wave. The characteristics of as many fictitious bodies as there are elementary waves to be combined are thus calculated. The list of the principal ones is as follows:

The semi-diurnal waves are first the mean lunar wave, which represents the mean movement of the semi-diurnal tide. This is the most important one of all, and the fictitious body that would engender it would have a mass equal to 95/100 of that of the Moon; secondly, the sidereal wave (lunar fraction) of period equal to a semi-sidereal day.' This gains on the first wave above, which is regulated by the lunar day, and the two coincide once in approximately fifteen days; thirdly, the lunar elliptical wave, which lags behind the first by a quantity equal to the mean anomalistic motion of the Moon in its orbit; the two coincide therefore at intervals of one month; fourthly, a mean solar wave; fifthly, a sidereal wave (solar fraction) ; and sixth a solar elliptical wave, the last three bearing the same relation to the Sun's action as the first three do to the Moon's action.

Next, we come to the group of diurnal waves, first, a lunar diurnal wave and, secondly, a sidereal diurnal wave (lunar fraction), thirdly, a solar diurnal wave and, fourthly, a sidereal diurnal wave (solar fraction) which represent almost exactly the effects of the respective declinations of the Moon and the Sun. These waves are, taken two by two, of almost equal magnitude, so that at the epochs of their coincidences the effect of each of the groups is sensibly doubled, while at the intermediate epochs the two nearly neutralise each other.

To these astronomical waves, we must add others of different characters. In the first place; there are meteorological waves, the result of regular diurnal and seasonal winds on the tides; these are inseparable from the diurnal and annual solar waves. Secondly, there are the waves resulting from the complications that the more or less irregular conformation of shores and depths impose on tidal phenomena. This is especially the case in estuaries where the tidal flow traverses large but shallow spaces.

Thus, each one of these waves (we may enumerate sixteen for the tides on the coasts of France, and twenty-one for the tides of the Indies) may be imagined to correspond to the action of a fictitious body, the characteristics of which can be calculated. It now remains to combine all these individual waves into one resultant whole. If this had to be done by calculation only, the work would be of great length and extreme difficulty, but Lord Kelvin originated the clever and simple idea of bringing the aid of a mechanical arrangement to the solution of the problem. He constructed an apparatus called the tide-predicter; when the particulars of each of the component waves are known for any given place such as, for example, Brest, the tide-predicter enables us to combine them all into a unique, complex, but mathematically exact curve which graphically represents the resultant phenomenon, that is to say, at the time represented by the abscissa of a point on this curve, the amplitude of the complex tide-wave which is the resultant of the composition of all the individual wave-elements is represented by the ordinate of that point.

The illustrious English physicist utilized, in the construction of this apparatus, the fact that the curves which express tidewaves are sine-curves, and that every sine-curve can be very easily traced by a pencil actuated by a rod and crank movement, the two trains of the crank and the revolving drum being connected (Fig. 28). If, therefore, we wish to compound a certain number of sine-curves each of which represents an elementary wave, say, for example, six, we take six crank rod systems (Fig. 29) ; the length of each crank represents the amplitude of the corresponding wave; the velocity of its rotation is governed by the period of the wave in question. Suitable trains of wheels en-able each to be rotated by a separate movement with its correct individual velocity. The position that each crank occupies in relation to that which represents the lunar, semi-diurnal wave when the latter is vertical shows the phase or relative displacement of the corresponding wave.

Each rod thus rises and falls, and if every one carried a pencil it would trace graphically on a paper, passing with a uniform motion, the sine-curve which is the graphical representation of the wave to which it corresponds. In order to combine all these movements to obtain the total effect, as regards both sign and magnitude, Lord Kelvin furnished each rod with a pulley. The rods are separated by spaces equal respectively to the diameters of the pulleys. One single thread passes through the grooves of all the pulleys; it is fixed at one of its ends and at the other end it carries a weight furnished with a pencil. When the whole system is started working it will be obvious that the weight, in rising and falling, represents at each instant the resultant ordinate of each of the partial ordinates of the six component sine-curves. If the pencil touches a cylinder covered with paper and turning uniformly, the required curve expressing the resultant phenomenon will be traced upon the paper. In reality there are sixteen pulleys in the apparatus in actual use for hydrographical purposes corresponding to sixteen component waves.

Such is the marvellous instrument which can do in a few minutes, exactly and without effort on our part, what would necessitate months of long calculations to achieve.

Before this apparatus was invented, and it dates back only a few years, tide almanacs, which are so necessary to sailors, had to be prepared in advance. At a given place the local circumstances, coasts, estuaries, the sea-bottom, etc., which so greatly influence the tides, are the constants for that place; the only variable quantities are the respective positions of the Earth, Moon, and Sun. Now every 18 years and 11 days these three bodies return to exactly the same relative positions. Consequently if the heights of the tides be observed at a given place during a period of i8 years and 11 days by means of the instruments called tide-recorders or tide-measurers, which are based on the principle of communicating vessels, we obtain the values of the tide for each day of the following period. This eighteen years and eleven days period was known to the ancients, who called it the Saros; seventy eclipses always occur during it, forty-one of which are of the Sun and twenty-nine of the Moon, and the eclipses observed during this period reoccur at corresponding epochs during the following period.

The tides propagate themselves in the form of a wave, and are consequently similar to the seismic waves of translation of which we have spoken in Chapter VII. Their velocity of propagation is, therefore, proportional to the square root of the depth of the water at the surface of which they are travelling. This explains a very curious observed fact: when the tide arrives at the west coasts of France the water rises slowly at first, from the level of low tide, but the velocity increases little by little and in proportion as the level rises the tide-stream is accelerated, the incoming current getting very strong and the rate of rise very rapid. This is a consequence of the above law of wave propagation, for the depth is nothing at first, at the time of low tide, and it increases in proportion as the sea recovers the sloping parts of the shore, being greatest just at the hour of high tide.

A similar consideration enables us to understand why there are at certain places, St. Malo, for instance, tides reaching a height of 14 metres, [46 ft.] while the theoretical tide ought not to surpass 6o centimetres [23.5 in.]. The explanation is that the tide manifests itself by an undulatory movement which communicates a considerable vibration velocity to the molecules of the water. So long as the wave travels over a deep ocean, for example, at the surface of depths of water of 4000, 5000, or even 6000 metres [2.5, 3, 3.75 miles] the velocity of propagation is sensibly constant. But, when the wave of the tide approaches the coasts of Europe it meets the continental plateau which is a kind of base or foundation on which the European continent stands, only 200 metres [66o ft.] below the surface of the sea. The force of the incoming tide is there-fore communicated to a much smaller liquid mass, and this results in the very high elevation of the water along the coasts, particularly in the English Channel, the narrow form of which accentuates the phenomenon. Maps may be made enabling us to follow the stages of the arrival of the tide-wave by the tracing of co-tidal lines joining the places where the wave arrives at the same time. The more the co-tidal lines are compressed together the greater the accentuation of the tides.

Two English hydrographers, Whewell and Lubbock, have carried still further this idea of the undulatory transmission of the tide. According to them, in order that the phenomenon of the tide-wave may be produced, it must take rise on an illimitable ocean in such a way that the wave following the movement of the attracting body can freely make the entire tour of the globe. Now these conditions are only realised in the southern ocean in the vast and terrible sea which completely surrounds the terrestrial globe and is bounded by the Antarctic continent, lying between this and Cape Horn, the Cape of Good Hope and Australia. Whewell and Lubbock believe that it is here that the generating tide-wave takes rise without any continental obstacle. The tide-wave which occurs in the Atlantic would, consequently, be only a secondary wave derived from the principal one.

Facts have been observed which give a strong support to this original idea. All along the Atlantic coasts, even on the extreme south coasts of the Argentine, are stations at which the tides are noted. It is, thus, possible to follow hour by hour the propagation of the northward travelling tide-wave. Now it has been shown that, when the tide-wave arrives, for example at midday, at the Straits of Magellan, it reaches Cape Corrientes near the mouth of the Rio de la Plata at midnight, that is, twelve hours later. After another twelve hours it gets to the Canary Isles and finally twelve hours later, that is to say at midnight on the second day, its influence is felt in the tide-recorder at Brest. It has, therefore, taken in all thirty-six hours to cross the Atlantic from south to north. It can also be proved that at Brest the equinoctial tide is only felt thirty-six hours after the theoretical moment when the Moon and Sun, whose attractive forces are then additive, produce the maximum possible tide. This is a very remarkable confirmation of Whewell and Lubbock's theory. Nevertheless, there is also a fact which runs counter to this theory. At no part in the islands of the Southern Sea, whether at Kerguelen, St. Paul, or New Amsterdam has the age of the tide been found to be nothing at all, as it should be in accordance with the preceding theory. Adhuc sub judice lis est.

Thus, the phenomenon of the tide is ruled, as regards its amplitude, by coastal configuration. In Europe, it is around England and France that the greatest variation of level takes place. Land-locked seas, on the other hand, such as the Mediterranean and the Baltic, have only insignificant tides; in the Gulf of Gabes tides of a metre [or yard] are sometimes observed and these arise chiefly from the Atlantic tide coming through the Strait of Gibraltar. On the other hand, in the partially landlocked seas and on large lakes, continuous variations of level are observed, the periodicity of which, although real, has no astronomical cause. These effects are readily visible on the Lake of Geneva. Their origin is probably meteorological ; when the atmospheric pressure distinctly increases at one end of an elongated lake, the level there is caused to fall and consequently that of the other end rises. Such a momentary inequality of the surface level leads to a re-establishment of hydrostatic equilibrium by means of a series of oscillations the duration of which depends on the size of the lake and its depth; theory and observation have always been in accordance on this point.

Seas such as the AEgean show also tide effects similar to that of the Lake of Geneva, and, at the epoch of the equinoxes, the joint effect of these and the small true tides which then occur is perceived. The variations of level have thus a complexity, more apparent than real, which the preceding considerations now enable us to elucidate completely.

The tides, as we have said, give rise to flow- or ebb-currents, according as the water is rising or falling. The configuration of the shore may be such as to be favourable to their establishment, or, on the other hand, it may tend to lessen them. In the former case, they may attain considerable strength and may present, at certain times, dangers for navigators. As cases in point we :have the race at Sein in Finisterre, and the Blanchard race in the English Channel, where during the equinoctial tides the velocity of the current exceeds eight miles per hour, and the whirlpools which occur in certain places after the change of the tide, on account of the meeting of two contrary currents, such as the Maelström in the north of Norway, Corryvrekan in the Hebrides, and the legendary whirlpool of Charybdis, more dangerous in the fable than in reality.

We shall now deal with other rhythmic movements exhibited by the waters of the sea, viz., the swell and the waves.

A representation on a small scale of the propagation of waves over the surface of water may be obtained by letting a pebble fall into a basin of water. Circular ripples or undulations are seen moving outwards from the point of immersion of the pebble towards the edge of the basin. The same thing occurs on a large scale at the surface of the oceans, which are always traversed by undulations of more or less importance. Such a movement is that produced in calm weather. When the wind begins to freshen and rise, the ridges of water which constitute the undulations of the swell lose their beautiful regularity; they cease to be symmetrical, becoming steep while their surfaces are covered with ripples and subsidiary wavelets. Little by little, under the influence of the wind, the wave slopes become hollowed and the summits begin to overhang; finally they give way and fall over, imprisoning a mass of air which escapes in bubbles of whitish foam, constituting the white-crested waves familiarly called "white horses." These are breaking waves.

The height of waves is sometimes considerable. While not reaching the values of 40 and 50 metres [130 to 150 ft.] which the assertions of ancient navigators attributed to them, on account of an optical error into which it is easy to fall, they do actually attain in the Southern Seas, at their highest, 15 to i6 metres [50 to 55 ft.], 10 to 11 metres [35 to 40 ft.] in the Indian Ocean, 8 to 9 metres [30 to 35 ft.] in the Atlantic, and, finally, in the Mediterranean 5 to 6 metres [15 to 20 ft.], always speaking of their highest, and, when freely propagated, far from the coasts.

For, when a system of undulation is not freely propagated but meets an obstacle, the phenomenon is complicated by that of interference between the direct movement, and the one reflected from the obstacle. In this way, the height of waves may become enormous. This is what happens when they beat upon the coasts; they rise up to heights of 40 to 50 metres [130 to 150 ft.] and fall back in masses of foam. The same thing happens when a large vessel is going at full speed in the opposite direction to the movement of propagation of the waves; these dash over the bow and may reach even to the highest superstructure when they sometimes do damage or wash away men.

When several series of waves, travelling in different directions, meet together, as a result of some special circumstances, interference phenomena are again produced, and the sea becomes agitated and choppy. This occurs at the centre of cyclones, where thousands of undulatory movements meet together, engendered by winds that have every possible direction since they form part of a whirling movement. In the Mediterranean, the closed contour of the coasts gives rise to reflected movements in all directions, and consequently the sea presents short and choppy waves which often render navigation difficult, although it is not actually very agitated.

The length of waves between consecutive summits is about 20 to 30 times their height; the great waves, 15 metres [5o ft.] in height, of the Southern Seas may thus reach lengths of 300 to 450 metres [325 to 500 yds.]. Consequently, the slope of these liquid heights is quite a gentle one, and this circumstance is a fortunate one, for, without this, the falling over of the crests when they break would render all navigation impossible.

If only a superficial examination be made, it seems that when undulations of water are caused in a basin, the water itself is transported towards the edge of the basin. More attentive observation will, however, show that this is not the case, for a small piece of wood thrown on the surface of the water rises and falls alternately with the passage of the waves, but does not move any nearer to the edge. The molecules of the liquid, therefore, move up and down in one place; two German physicists, the brothers Weber, have experimentally studied the matter and have found as a result that each aqueous molecule describes a closed curve (Fig. 30), and that it is the combination of these vibratory movements, transmitted from molecule to molecule which constitutes the propagation of undulatory movements. In proportion as the molecule in question lies deeper under the water, the circular path described by it is flatter and so becomes a more and more flattened ellipse which, ultimately, is reduced to a straight line. At great depths, therefore, wave propagation takes place by a simple rectilinear horizontal movement, for-ward and backward, of the molecules of the liquid. Experiment has shown that surface agitations make themselves felt down to a depth of 300 to 35o times the height of the undulations produced; there is, thus, a level below which superficial agitation is practically not transmitted at all.

The most simple manifestation of undulatory movement at the surface of the sea is the swell characterised by the absence of that white foam which sailors call "white horses." It forms at the surface of the sea regular ridges, with regularly curved sides and which move majestically over the water when the atmosphere is calm.

The undulation of the swell is in profile the curve above represented and' is called by mathematicians a cycloid. Such an undulation is characterised by its length, by which is meant the constant distance between two consecutive crests; by its velocity of propagation, which is the distance traversed in a second by the condition of undulation ; by the period, which is the time taken for one crest to succeed the next, and, finally, by the amplitude or height of the undulations, that is to say, by the vertical distance between the crest and the hollow of the wave.

The movements of the sea represent the production of a considerable sum of mechanical energy.

Considering first the waves, their velocity of propagation is about twenty-five marine miles per hour, that is, more than 45 kilometres [28 standard miles]. Furthermore, a large mass of water is contained in a wave io metres [35 ft.], high, and, as the energy of the wave is its mass multiplied by the square of the velocity, it so attains a considerable amount. A wave of the height and velocity just mentioned develops about two thousand horse-power per metre [or yard] width. Stephenson has also measured directly the force exerted on a given surface by the shock of such waves and finds it thirty tons per square metre [French ton =2204.62 lbs.]. If we remember that a like force is produced every ten or fifteen seconds, this being approximately the period of these waves, for periods of some days, it is obvious that they represent a large amount of energy.

These powerful masses of water beat against the coasts, and by long-continued attack wear away the rocks and break off portions of them. In this way the granites of Brittany are scooped and hollowed, and the chalk cliffs of Normandy are undermined from their base. Similarly defensive works and breakwaters that man has erected, at the cost of years of difficult labour, to protect harbours may sometimes be broken through and destroyed by a single storm in a few minutes. Perhaps, at some future time, it may be possible to harness these hitherto unutilised forces; then transmitted inland by means of electric currents, the movement of the waters of the sea could be put to use instead of, as at present, only causing destructive effects.

As regards the tides, the power represented by the alternate rising and falling of the level of the sea is also considerable, and would be easier to utilise; it would suffice to construct vast basins which could actually be done in many cases by closing in an estuary by means of a dam, forming a natural reservoir which would be filled at high tide by the flowing in of the sea. The opening could then be shut, and the water thus maintained at the high level would work turbines in rushing down when the level of the water outside had fallen; consequently the power would be available. In regions such as that of St. Malo, where the tides reach a height of 15 metres [5o ft.] at the equinoxes there would be an ample reserve of energy. In the bay of Mont St.-Michel each square kilometre [.38 sq. mile] of the sea surface represents an average force of 20,000 horse-power, and the bay is not less than 300 square kilometres [116 sq. miles] in area. If, therefore, it was closed by an embankment we should have available about 6,000,000 horse-power, and the work would not be more difficult than the making of the Suez Canal or of a railway across the Sahara. And a number of other bays would lend themselves to similar procedure. The damming of the Rance would also give more than 200,000 available horsepower.