ON WAVES. 389 



and W 2 w represents one-tenth of an inch in height of the wave ; so that 

 the height of the wave is indicated to the eye by the number of lines. 

 These observations are made on concentric circles. 

 In figs. 1 and 2 the same kind of observations is represented, only along 

 straight lines. 



Plate LV. 



Waves of the Second Order. — Standing Waves in Running Water. 



The forms of the waves in these figures are the same as those in figs. 9, 10 

 of Plate LIL, being all cycloidal ; with this difference only, that the waves 

 in Plate LII. were moving along the standing water with a uniform velo- 

 city, while those in Plate LV. are standing in the running water. The ge- 

 nerating course in this case is a large obstacle or large stone in the running 

 stream. On this the water impinges ; it is heaped up behind it ; it acquires 

 a circular motion which is alternately coincident with and opposed to the 

 stream ; the water having once acquired this circular oscillating motion in 

 a vertical direction retains it, the water is alternately accumulated and ac- 

 celerated, and thus standing waves are formed, as shown in figs. 1 and 2. 



Figs. 3 and 4 exhibit a remarkable case of the coexistence in one stream of 

 two sets of waves moving with velocities differing in about the proportion 

 of two to three. On one side of a stream there projected a ledge of rock 

 M, over which fell a thin sheet of water into a large pool, nearly still, 

 without generating any sensible wave. On the opposite side a deep violent 

 current was running round the obstacle with great rapidity. The middle 

 part of the channel was occupied by a large boulder, over which also a 

 stream flowed, generating standing waves with a smaller velocity. These 

 waves are also remarkable for non-diffusion, as they will preserve their 

 visible identity to a great distance without being dissipated. 



Plate LVI. 



Waves of the Second Order Their Mechanism. 



All the waves of the second order, whether standing waves in running 

 water or travelling waves in standing water, exhibit the forms of the curves 

 B AB C D in fig. 1. These are cycloids, having for their base the rec- 

 tilineal distance A C, and for their height the corresponding circles. In 

 the case of standing waves in running water these cycloids represent the 

 actual paths of individual particles of water in the running stream, as shown 

 in Plate LV. In the case of travelling waves in standing water, the circles 

 represent the paths described by the individual particles of water, and the 

 cycloids the visible moving surface presented to the eye. The motion of 

 oscillation in the upper half of the circle is in running water, opposite to 

 the motion of the stream, and in standing water is in the same direction as 

 the visible motion of transmission of the waves. The figure shows the rapid 

 diminution of the motion of oscillation with the depth. I am indebted for 

 this figure to M. Gerstner, whose theory it illustrates, and I have given it 

 because I find it represent my own observations as correctly as any figure 

 of my own could do. I have only found it necessary in reconstructing his 

 figure to clear it of some slight inaccuracies. The shaded parts on the left 

 show the different forms which given portions of water successively assume 

 during wave motion. The circular orbits are divided into equal portions, 

 numbered 1, 2, 3, 4, &c., to show that the particles of water are in those 

 points of the circles at the same instants the corresponding particles are at 

 the points 1, 2, 3, 4, &c. of the cycloidal paths. 



