If it be assumed that x n cos (a t — nix) + sin (tit —m.r), n and a being functions of y, the above equations of continuity and of equal pressure give, on the supposition that gravity is constant, that no extraneous forces act, and retaining for the present only the first dx power of , or of the horizontal displacement sPx d?x dye dsi ". 0.
From these two equations are obtained the values of x and T in terms of A cos (at —m-e) and n sin (iit These values will not be altered if ins is increased or diminished by I one, two, three. &c. whole circumferences, that is, if x is increased or 4 w diminished by — 2w — while I remains the same; therefore — m 2w is the value of the increments of x which correspond to points whero the particles of water are in the same condition with respect to dis truhanoo, that is, is the length of a wave. Again, the values will not be altered if nt is increased or diminished by whole circumferences, that is, if I is increased or diminished by 2w 4w, fie., while s remains 'Sr tho same; therefore -; is the increment of time which corresponds to the particlea of water being successively in the like state of disturb 2w rune, that is — is the period of a wave, or the time between two successive formations of a wave-summit at the same place. Therefore is the velocity of the wave ; and, from the value found for it by the sa theory, it follows that tho velocity depends on in and on the depth of the water : the latter being constant, the velocity depends on the length of the wave, or it depends on the time in which a particle of water makes a complete vibration. If the length of a wave or the time of its vibration is given, the velocity will vary with the depth of the water.
From a table of the computed velocities of waves of different lengths, and with different depths of water, it is found that when the length of the wave is not greater than the depth of the water, the velocity of the wave is proportional to the square root of its length : also when the length is not less than one thousand times the depth of water, the velocity is proportional to the square root of the depth, and is the same as that which a body would acquire in falling from rest through a height equal to half that depth. The greatest horizontal and vertical displacement of a particle being computed for different values of the length of the wave and the depth of the water, it appears that when the latter is great, compared with the former, as in the open sea, the motion of the water far below the surface is very small compared with the motion at the surface, and at a depth equal to the length of wave it is only about of the motion at the surface. On the same suppo sition, the greatest horizontal motion is equal to the greatest vertical motion. When the length of the wave is great compared with the depth of the water, as in tide-waves, the horizontal motion of the particles is nearly the same from the surface to the bottom, and the vertical motion varies with the distance from the On the same supposition, the vertical motion of the superior particles is much less than their horizontal motion.
The movement of a particle of water near the surface may he deter mined from the values given by the theory to x and Y : if the waves are small, so that A may ho considered as equal to n, we have i (xi +V) t= o, a constant; which, being the equation of a circle, it follows that the particles move in the circumference of a circle whose radius is ; hut if the length of the wave is great compared with the depth of water, the equation is that of an ellipse. These last deductions from the theory are conformable to what has been observed in experi mental waves, as above mentioned. It follows that, in a long tide-wave flowing up a channel, the horizontal velocity iu the direction of the wave's motion is the greatest at the summit of the wave—that is, at high-water : at the place of greatest depression—that is, at low-water ' —the motion is most rapid downwards; and at the mean level the water is for a time stationary.
In investigating theoretically the phenomena of waves by whatever cause produced, if the lengths of the waves are very great compared with the depth of the canal in which they move, it becomes necessary to retain the second and even higher powers of or of the horizontal dx displacement, in the equations of continuity and of equal pressure; but the vertical oscillations being then small, the value of (I' Y may be neglected.* Then, if the perturhating actions of the sun and moon ere not considered, the integration of the differential equation of equal pressure gives a value of the vertical displacement at the surface, or the height of the wave above the mean elevation, in terms which contain k sin (nt —nix) and kx sin (2n 4-2 nix), k being the depth of water in the canal. Tracing an undulating lino whose ordinates are the values of that vertical height, corresponding to different values of x, the horizontal distance from the mouth of the canal, which is sup posed to open to the sea; it is found that, near the opening, the front and rear slopes of the waves are of equal lengths and of similar forms ; but as the distance from the sea becomes greater, the front slope is shorter and steeper, and the rear slope longer and more gentle. At a great distance the latter becomes nearly horizontal in the middle, and at length it divides into two parts, so that the wave becomes double. Near the sea, also, the time occupied by the rise of the wave is equal to the time occupied by its descent : at a certain distance the rise takes place in loss time than the descent ; and at a still greater distance the descent, after having been rapid, is checked, or changed into a rise, to which another rapid descent succeeds ; so that there seem to be two tides, or elevations of the water, in the upper part of the canal, corresponding to one elevation at the mouth.