The value t or the velocity of the particles of water, is found also to contain the sines and cosines of the angles above mentioned ; and, substituting in these the greatest positive and the greatest negative values of the elevation, it is found that the velocity corre sponding to the first of these values—that is, at the top of the wave— is less than the velocity corresponding to the other ; but the motion, - in the first case, is up the canal, and in the other down it, and these are nearly the same as the greatest velocities of the water : conse quently, the velocity of the flow of the wave in the canal ie less than that of the ebb. The preceding conclusions relate to the case in which the water was at rest in the canal previously to the formation of the wave : in the event of the water having a general movement towards the sea, the time in which the wave rises, or the time from low-water to high-water, is still less than the time of the descent; but the difference between the two times is greater than in the former case.
If a section of the bed of the canal, instead of being rectangular, has the form of an isosceles triangle, the investigations show that the velocity of the wave would be equal to that of a wave in a rectangular bed whose depth is equal to half the perpendicular of the triangle. If the section were a parabola, the velocity would be two-thirds of that which the waves would have in a rectangular bed of equal breadth and depth.
When the water, still supposed to be in a canal of uniform breadth and depth, is disturbed by extraneous forces, as the attraction of the sun or moon, the term p in the equation of equal pressure is conceived to consist of two, one represented by it sin. (it—mx) for the horizontal intensity of such force in the direction of x, and the other by a cos. (it—mx) for the vertical intensity; and the equation for being then satisfied by the equation x=¢" (y) sin. (it—mx), in which (p" (y) represents the second differential coefficient of a function of y, there is obtained a value of x at the surface of the fluid in terms of sin. (it— mz), and a value of the height above the level of still water In terms of cos. (it—mx). The wave thus indicated depends upon the continuance of the actions of the extraneous disturbing forces, and is designated by Mr. Airy the forced tide-wave. This wave, he observes, would cease to exist if those forces were to cease; but other waves resulting from the previous action would still continue to exist, and these he distinguishes by the name of free tide-waves. If the canal be supposed to surround the earth at the equator, the length of the forced tide-wave is equal to half the circumference of that great circle ; and from the expressions for x and v, it appears that the effect of the vertical disturbing forces on the phenomena of the tides is insignificant, almost the whole sensible effect being due to the horizontal force.
Taking into account the effects of friction, which may be considered as a horizontal retarding force proportional to the velocity, and which may consequently be represented by — f x the value of x contains terms involving the sines and cosines of angles represented by it —mx and it + 2.r, and the expression for the vertical elevation contains the sine of it—nix. The analytical expression arising from the introduction of this additional perturbation indicates the fact that the highest tides take place later than the times at which the disturb ing forces arising from the action of the sun or moon are the greatest; and this circumstance gives to the wave theory an important advantage over those of Newton and La Place ; for in both these theories the greatest tides take place when the force is the greatest.
In the case of a canal bounded at both extremities, the expression for x, the horizontal disturbance of a particle, is found to consist of two parts, one of which is the horizontal movement due to the disturb ing forces, and the other a combination of free tide-waves, probably caused by reflexions of the forced tide-waves from the opposite ends of the canal. When a canal so bounded is of small extent, the horizontal motion of the particles is found to be the greatest in the middle of its length, and to diminish gradually to the ends, where it vanishes. There is proved to be no variation of level in the middle of the length, and the variation in other parts is proportional to the distance from the middle, the elevation at ono end taking place at the same time as the depression at the other. It results, also, that the greatest horizontal stud vertical displacements of the particles take place at the same time ; whereas in other circumstances, from the circular or elliptical motions of the particles, the greatest horizontal displacements take place when the vertical displacements are zero, and vice versa.
In a deep gulf open to the sea at one end and closed at the other, and in which the waters have a tidal fluctuation, the termination of the flow upwards takes place at the mouth a considerable time after high-water ; but near the bottom of the gulf the difference between the times is very small. When a tide-wave is propagated up a river, the analysis shows that the vertical elevations of the wave, and also the horizontal motion of the particles of water, diminish continually as the wave advances : also the direction of the tide-current changes sooner after the instant of high-water than it would if friction were not con sidered. When a river runs on a declivity towards the sea, the latter being affected by tides, it is shown that the low-water at certain points up the river may be higher than the level of high-water on the sea.