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deed, if so heavy a body as water were to move at the rate of 1000 miles an hour, it would cause universal destruction, since in the most violent hurricanes the velocity of the wind is little more than 100 miles an hour. Besides, it is evident that no ship could either sail or steam against it. When the water is shallow, however, there is a motion of translation in the water along with the tide.
In the deep ocean the undulating motion consists of two distinct things—an advancing form and a molecular movement. The motion of each particle of water is in an ellipse lying wholly in the vertical plane; so that, after the momentary displacement during the passage of the wave, they return to their places again. The resistance of the sea-bed is insensible in deep water; but when the tidal wave, which extends to the very bottom of the ocean, comes into shallow water with diminished velocity, the particles of water moving in vertical ellipses strike the bottom, and by reaction the wave rises higher; and that being continually repeated, as the form moves on the wave rises higher and higher, bends more and more forward, till at last it loses its equilibrium, and then both form and water roll to the shore, and the elliptical trajectories of the particles, which in deep water were vertical, incline more and more, till at length they become horizontal. The distance from the shore at which the water begins to be translated depends upon the depth, the nature of the coast, and the form of the shore. Mr. Scott Russell has demonstrated that in shallow water the velocity of the wave is equal to that which a heavy body falling freely by its gravity would acquire in descending through half the depth of the fluid.
It is proved by daily experience, as well as by strict mathematical reasoning, that, if a number of waves or oscillations be excited in a fluid by different forces, each pursues its course and has its effect independently of the rest. Now, in the tides there are three kinds of oscillations, depending on different causes,
and producing their effects independently of each other, which may therefore be estimated separately. The oscillations of the first kind, which are very small, are independent of the rotation of the earth, and, as they depend upon the motion of the disturbing body in its orbit, they are of long periods. The second kind of oscillations depend upon the rotation of the earth, therefore their period is nearly a day,' The oscillations of the third kind vary
with an angle equal to twice the angular rotation of the earth, and consequently happen twice in twenty-four hours (N. 157). The first afford no particular interest, and are extremely small; but the difference of two consecutive tides depends upon the second. At the time of the solstices this difference, which ought to be very great according to Newton's theory, is hardly sensible on our shores. La Place has shown that the discrepancy arises from the depth of the sea, and that if the depth were uniform there would be no difference in the consecutive tides but that which is occasioned by local circumstances. It follows, there fore, that, as this difference is extremely small, the sea, considered in a large extent, must be nearly of uniform depth, that is to say, there is a certain mean depth from which the deviation is not great. The mean depth of the Pacific Ocean is supposed to be about four or five miles, that of the Atlantic only three or four, which, however, is mere conjecture. Possibly the great extent and uniformly small depth of the Atlantic over the telegraphic platform may prevent the difference of the oscillations in question from being perceptible on our shores. From the formulæ which determine the difference of these consecutive tides it is proved that the precession of the equinoxes and the nutation of the earth's axis are the same as the sea fornied one solid mass with the earth.
The oscillations of the third kind are the semi-diurnal tides so remarkable on our coasts. In these there are two phenomena particularly to be distinguished, one occurring twice in a month, the other twice in a year.
The first phenomenon is, that the tides are much increased in the syzigies (N. 158), or at the time of new and full moon : in both cases the sun and moon are in the same meridian ; for when the moon is new they are in conjunction, and when she is full they are in opposition. In each of these positions their action is combined to produce the highest or spring tides under that meridian, and the lowest in those points that are 90° distant. It is observed that the higher the sea rises in full tide, the lower it is in the ebb. The neap tides take place when the moon is in quadrature. They neither rise so high nor sink so low as the spring tides. It is evident that the spring tides must happen twice in a month, since in that time the moon is once new and once full. Theory proves that each partial tide increases as the cube of the parallax or apparent diameter of the body producing it, for the greater the apparent diameter the nearer the body and the more intense its action upon the sea; hence the spring tides are much increased when the moon is in perigee, for then she is nearest to the earth.
The second phenomenon in the tides is the augmentation occurring at the time of the equinoxes, when the sun's declination is zero (N. 159), which happens twice in every year. The spring tides which take place at that time are often much increased by the equinoctial gales, and, on the hypothesis of the whole earth covered by the ocean, would be the greatest possible if the line of the moon's nodes coincided with that of her perigee, for then the whole action of the luminaries would be in the plane of the equator. But since the Antarctic Ocean is the source of the tides, it is evident that the spring tide must be greatest when the moon is in perigee, and when both luminaries have their highest southern declination, for then they act most directly upon the great circuit of the south polar seas.
The sun and moon are continually making the circuit of the heavens at different distances from the plane of the equator, on account of the obliquity of the ecliptic and the inclination of the lunar orbit. The moon takes about 292 days to vary through all her declinations, which sometimes extend 2880 on each side of the equator, while the sun requires nearly 3654 days to accomplish his motions through 231° on each side of the same plane, so that their combined action causes great variations in the tides. Both the height and time of high water are perpetually changing, and, although the problem does not admit of a general solution, it is necessary to analyse the phenomena which ought to arise from the attraction of the sun and moon, but the result must be corrected in each particular case for local circumstances, so that the theory of the tides in each port becomes really a matter of experiment, and can only be determined by means of a vast number of observations, including many revolutions of the moon's nodes.
The mean height of the tides will be increased by a very small quantity for ages to come, in consequence of the decrease in the mean distance of the moon from the earth; the contrary effect will take place after that period has elapsed, and the moon's mean distance begins to increase again, which it will continue to
do for many ages. Thus the mean distance of the moon and the consequent minute increase in the height of the tides will oscillate between fixed limits for ever.
The height to which the tides rise is much greater in narrow channels than in the open sea, on account of the obstructions they meet with. The sea is so pent up in the British Channel that the tides sometimes rise as much as fifty feet at St. Malo, on the coast of France; whereas on the shores of some of the South Sea islands, near the centre of the Pacific, they do not exceed one or two feet. The winds have great influence on the height of the tides, according as they conspire with or oppose them. But the actual effect of the wind in exciting the waves of the ocean extends very little below the surface. Even in the most violent storms the water is probably calm at the depth of ninety or a hundred fathoms. The tidal wave of the ocean does not reach the Mediterranean nor the Baltic, partly from their position and partly from the narrowness of the Straits of Gibraltar and of the Categat, but it is very perceptible in the Red Sea and in Hudson's Bay. The ebb and flow of the sea are perceptible in rivers to a very great distance from their estuaries. In the Narrows of Pauxis, in the river of the Amazons, more than five hundred miles from the sea, the tides are evident. It requires so many days for the tide to ascend this mighty stream, that the returning tides meet a succession of those which are coming up; so that every possible variety occurs at some part or other of its shores, both as to magnitude and time. It requires a very wide expanse of water to accumulate the impulse of the sun and moon, so as to render their influence sensible ; on that account the tides in the Mediterranean and Black Sea are scarcely perceptible.
These perpetual commotions in the waters are occasioned by forces that bear a very small proportion to terrestrial gravitation : the sun's action in raising the ocean is only the 38448000 of gravitation at the earth's surface, and the action of the moon is little more than twice as much ; these forces being in the ratio of 1 to 2:35333, when the sun and moon are at their mean distances from the earth. From this ratio the mass of the moon is found to be only the is part of that of the earth. Had the action of the sun on the .ocean been exactly equal to that of the moon, there would have been no neap tides, and the spring tides would have been of twice the height which the action of either the sun or
moon would have produced separately—a phenomenon depending upon the interference of the waves or undulations.
A stone plunged into a pool of still water occasions a series of waves to advance along the surface, though the water itself is not carried forward, but only rises into heights and sinks into hollows, each portion of the surface being elevated and depressed in its turn. Another stone of the same size, thrown into the water near the first, will occasion a similar set of undulations, Then, if an equal and similar wave from each stone arrive at the same spot at the same time, so that the elevation of the one exactly coincides with the elevation of the other, their united effect will produce a wave twice the size of either. But, if one wave precede the other by exactly half an undulation, the elevation of the one will coincide with the hollow of the other, and the hollow of the one with the elevation of the other; and the waves will so entirely obliterate one another, that the surface of the water will remain smooth and level. Hence, if the length of each wave be represented by 1, they will destroy one another at intervals of 1, 1, 1, &c., and will combine their effects at the intervals 1, 2, 3, &c. It will be found according to this principle, when still water is disturbed by the fall of two equal stones, that there are certain lines on its surface of a hyperbolic form, where the water is smooth in consequence of the waves obliterating each other, and that the elevation of the water in the adjacent parts corresponds to both the waves united (N. 160). Now, in the spring and neap tides arising from the combination of the simple solilunar waves, the spring tide is the joint result of the combination when they coincide in time and place; and the neap tide happens when they succeed each other by half an interval, so as to leave only the effect of their difference sensible. It is, therefore, evident that, if the solar and lunar tides were of the same height, there would be no difference, consequently no neap tides, and the spring tides would be twice as high as either separately. In the port of Batsha, in Tonquin, where the tides arrive by two channels of lengths corresponding to half an interval, there is neither high nor low water on account of the interference of the waves.
The initial state of the ocean has no influence on the tides; for, whatever its primitive conditions may have been, they must soon have vanished by the friction and mobility of the fluid. One of the most remarkable circumstances in the theory of the tides is