1. The existence of a great primary wave of fluid, differing in its origin, its phenomena, and its laws, from the undulatory and oscillatory waves which alone had been investigated previous to the researches of Mr. Russell, have been confirmed and established. This wave was first observed by him iu 1834.
2. The velocity of this wave in channels of uniform depth is hide pendent of the breadth of the fluid, and equal to tho velocity acquired by a heavy body falling freely by gravity through a height equal to half the depth of the fluid, reckoned from the top of the wave to the bottom of the channel.
3. The velocity of this primary wave is not affected by the velocity of impulse with which the wave has been originally generated, neither does its form or velocity appear to be derived in any way from the form of the generating body.
4. This wave has been found to differ from every other species of wave in the motion which is given to the individual particles of the fluid through which the wave is propagated. By the transit of the Native the panicles of the fluid are raised from their places, transferred forward in the direction of the motion of the wave. and permanently deposited nt test in a new place at a considerable distance from their original position. There is no retrogradation, no oscillation ; the motion is all in the same direction, and the extent of the transfer ence is equal throughout the whole depth. Hence this wave may be descriptively designated the great primary wave of translation. The motion of translation commences when the anterior surface of the wave is vertically over a given series of particles; it increases in velocity until the crest of the wave has come to be vertically above them, and from this moment the motion of translation is retarded, and the particles are left in a condition of perfect rest at the instant when the posterior surface of the wave has terminated its transit through the vertical plane in which they lie. This phenomenon has been verified up to the depth of 5 feet.
5. The elementary form of the wave is eyeloidal; when the height of the wave is small in proportion to its length the curve is the pro late cycloid, and as the height of the wave increases the form approaches that of the common cycloid, becoming more and more cusped until at last it becomes exactly that of the common cycloid, with a cusped summit ; and if by Any means the height be increased beyond this, the curve becomes the curtate cycloid, the summit assumes a form of unstable equilibrium, tot ters, and falling over on one side forms a crested wave or breaking surge.
6. A wave is possible in forms of channel where the depth is not uniform throughout the whole depth. It appears however that where the difference between the depth of the sides is considerable, one part of the wave will continue during the whole period of propagation in the act of breaking, so as to show that in these circumstances a con tinuous wave is impossible. In other cases the ridge of the wave rises so much higher on the shallower part of the fluid as to produce a given velocity without exceeding the limits of equilibrium, and in those cases the wave becomes possible, and the velocity appears to coincide closely with that which we obtain by supposing the wave resolved into vertical elements, each having the velocity due to the depth, and thou integrating. It results that : e In the rectangular channel the velocity is that of gravity duo to half the depth.
In the sloping or triangular channel the velocity is that due to one third of the greatest depth.
In a parabolic channel the velocity is that due to three-eighths or three-tenths of the greatest depth, according as the channel is convex Jr concave.
The velocity of the great primary wave of translation of a fluid is that due to gravity acting through a height equal to the depth of the centre of gravity of the transverse section of the channel below the surface of the fluid.
7. The height of a wave may be indefinitely increased by propaga tion into a channel which becomes narrower in the form of a wedge, the increased height being nearly in the inverse ratio of the square root of the breadth.
8. If waves be propagated in a channel whose depth diminishes uniformly, the waves will break when their height above the surface of the level fluid becomes equal to the depth at the bottom below the surface.