It must not be imagined that when water is agitated by waves, its whole masa has the movement which at first eight appears from the observed progressive motion of the undulation ; and, in order to account for the formation and motion of waves, it is sufficient to assume that the particles of water, when disturbed, have merely small oscilla tory motions in horizontal and vertical directions. When, from any cause, as the fall of a stone into it, the water becomes agitated, a series of horizontal motions to and fro aro produced ; and while in a slender vertical column of water these motions arc equal and in one direction, the surface neither rises nor falls; but if, in two neighbouring columns, the particles advance to meet each other, the water becoming com pressed, the surface rises; If the particles recede from one another, those above descending by gravity, the surface falls. These different horizontal movements existing successively in the same vertical column, and rihnultaneously in thou which arc adjacent to each other, the anr face of the water becomes undulated. In order, however, to under stand the true movements of waves, let the straight lino a k represent the surface of water when undisturbed, and, disregarding tho hori zontal oscillations by which the water is alternately compressed and dilated, let the particles be conceived to ascend and descend alternately in vertical lines, that is, in lines parallel to a' a am, which is supposed to be perpendicular to a k. Now, at a given instant let the surface of the water have, in a vertical plane, the form &c., and let the force of ascent cause the particles in the line a b m to be raised up to the line a b' m in a portion of time represented by that force becom ing less as it is farther from a horizontally, and ceasing at m. At this place the force of descent commencing, the particles in the line mcdn fall simultaneously with the rise of the particles in a b m, and at the end of the same time T, they occupy the line sr c'd'u. Here the force of ascent acts, and the particles in a e fp at the end ef the same time occupy the line ne'fp, and so on. Thus at the end of the time T the surface of the water has assumed the form a'b'e'd', &c. After this time the force of descent on the particles in the line a'm' causes those par ticles to fall vertically, during a time equal to T, into the line am ; at m' that farce ceases, and the force of ascent raises the particles in m'b'c'n vertically into the line m'b"cn', and so on ; thus, at tho end of the time 2T from the given instant the surface of the water has the form ab"cd", &c. In like manner, at the end of,the time 3 T the farces of descent and ascent will have brought the particles into the line an'e'd', ; and at the end of the time 4 T the particles will he again in the line abed, &c.: so that in this time every particle of fluid has made one complete vibration vertically, as an'aa'''n, and within the Fame time the top of the wave has assumed successively the positions d, e', f', rf" , h. The horizontal distance from d to Jr is called the length of a wave ; let it be represented by a, and let T express the time 4 T in which the summit of a wave has passed from d to It ; then .11 is called the velocity of the wave. On observing the characters of experimental waves in troughs with glass sides, it is found that, by the combinations of the horizontal and vertical vibrations, the particles of water describe the peripheries of circles or ellipses. In the upper parts of the curves, nee; the tops of the waves, the particles move with their greatest velocitiea in the direction in which the wave is advancing ; in the lower parts, near the bottoms of the waves, they are moving with their greatest velocities backwards ; and at the extremities ef the horizontal diameters, about the level of the water's surface when at rest, the motion is almost wholly vertical.
The varying attraction of the sun or moon on the particles of water in the ocean is alone sufficient to produce the perturbations by which waves are formed ; snd if it be assumed that the solid nucleus of the earth is covered entirely with water, both nucleus and water being originally spherical, those perturbations will bring the surface of the water to a spheroidal form, the longer axis being in the direction of a line joining the centres of the earth and luminary ; there will consequently exist, at the same instant, two great waves whose summits are at a distance from one another equal to half the circumference of the earth. But the motion of the water in the tide-wave is totally
unlike that in an ordinary snrface-wave such as the wind produces ; and while the latter, even in the most violent storms, agitates the sea fo a very trifling depth, the tide-wave affects the whole depth of the ocean equally, from the bottom to the surface.
Very little attention to the phenomena of the tides suffices to show that, in situations where the recurrences of high-water are nearly regular, the greatest elevation of the water takes place at intervals of about 12 hours 25 minutes, and the greatest depressions at the like intervals of time from each other ; each greatest depression taking place about 6 hours 12 minutes after the instant of greatest elevation. Now the interval between two successive culminations of the moon on the same side of the geographical meridian of any place varies from about 24 hours 40 minutes to 25 hours; and thus the intervals between the times of high-tide have evidently a connection with the diurnal revolution of the moon; moreover, the occurrence of high-water at any place is observed to have a dependence on the position of the moon with respect to the meridian of the place ; at a few ports it coincides with the time that the moon is on meridian, but in general it takes place some time before or after the culmination. The position of the moon at the time is, however, subject to certain variations even at the same port ; and it differs considerably at different places. The eleva tions also of the water with respect to the mean level differ; in some places, during about half the year, the high-tide which occurs when the moon is above the horizon is greater than that which occurs when the moon is below, and during the other half-year the phenomenon is reversed. In every place, at about the times of new and full moon, the high-tides attain their greatest elevation ; and at about the times of the quadraturcs, the least: the former are called spring-tidxs, and the latter neap-tidca.
In bays and harbours, the time of high-water coincides with that at which the current ceases to flow, but this is not the ease with the seas which communicate at both extremities with the ocean. For, if it be imagined that a tide-wave flows in at one of the extremities, this will cause an elevation of the waters ; but the waters which are passing off at the opposite extremity cause, at the same time, a depression, or, at least, a diminution of that elevation ; the surface, therefore, moat ho the highest when the current flows with equal rapidity at both ex tremities, and not at the moment preceding tho turn of the tide.
When the stream continues to flow up for three hours after it is high water, it is said to make tide and half-tide ; if it continues to flow during one hour and a half, it is said to make tide and quarter-tide, and so on. Near the shores of the British Channel, probably in con sequence of the obstructions caused by the land, or the disturbances at the mouths of rivers, the progressive movement of the tide-wave is more retarded than in the middle ; and in some places the current has curvilinear motions, which on the French and English sides are in opposite directions. The race of Portland is a current produced by the tide-wave, while advancing along the shore; being arrested by the promontory till it attains a height which alrowsit to flow off obliquely considerable velocity.