ON WAVES. 387 



drawn horizontal lines, which are intersected by vertical lines from each of 

 the divisions of the straight line dd, as shown in the figure. A continuous 

 line, passing through these points of intersection, has for its vertical ordi- 

 nates the versed sines of the arcs of the circle, while its abs-^issee are pro- 

 portional to the arcs themselves. Such a line is the curve of versed sines, 

 and gives a first approximation to the form of the wave of the first order. 

 Fig. 4 gives a second approximation to the form and the representation of 

 the pheenomena of the wave of the first order. A D fZ is taken equal to 

 the length of the wave in the first approximation = 6*28 times the depth 

 of the fluid in repose ; on dc =^ the height of the wave, a circle is de- 

 scribed and divided into equal arcs as formerly, and thus a dotted line, 

 A C c?, is formed as before, being the first approximation to the wave form. 

 These equal arcs being taken to represent equal times, the versed sines 

 also represent the rise and fall of the surface of the wave during equal 

 successive intervals of time. But hitherto we have neglected the motion 

 of translation, the horizontal transference of each vertical column of fluid 

 in the direction of wave transmission simultaneous with the vertical motion. 



y 

 Take the length A to A', such that A A' x A B shall = _ = the volume 



of water generating the wave divided by the breadth of the fluid. This 

 length, A B, in a small wave will be about three times the height of the 

 wave. Take A A' as the major axis of an ellipse, of which the minor axis 

 is C D or c d, the height of the wave. Let the horizontal lines through 

 the equal arcs of the small circle c d he extended to pass through the el- 

 lipse A A', and from the points of division let fall perpendiculars on A A' 

 on the points 1, 2, 3, 4, 5, 6, 7, 8, 9, then the lines on the axis A A', viz. 

 A 1, A 2, A 3, A 4, A 5, A 6, A 7, A 8, A 9, A A' represent the amount 

 of horizontal transference effected during the same time, in which a given 

 particle on the surface is rising and falling through the versed sines of the 

 equal arcs, viz. d\, dO,, d3, dA:, d 5, d6, d7, d8, d9, dd. Let us now 

 effect this horizontal transference on each point of the surface on the first 

 wave A C d, by advancing the point 1 horizontally through a distance 

 equal to A 1 ; 2 through a distance A 2 ; 3 through a distance a 3, and so 

 on, and we shall get a curve A' C d, which closely represents the form of 

 the wave, and also its phasnomena of horizontal translation = throughout 

 the whole depth to A 1, A 2, A 3, A 4, A 5, &c. 



Fig. 5 is obtained in the same way as fig. 4, only for a larger wave ; where 

 the height is nearly equal to the depth of the fluid, the ellipse is nearly a 

 semicircle. The same ellipse represents also the absolute path of a particle 

 on the surface during wave transmission. Ellipses of the same major axes, 

 but having their minor axes diminishing with their distance from the bot- 

 tom of the channel, will represent the simultaneous motions of particles 

 below the surface. 



Fig. 6 shows a single particle path, and three successive positions of the 

 wave outline in regard to it. The figures 1, 2, 3, 4, 5, &c., give the si- 

 multaneous positions of the particle referred to the wave surface, and the 

 same particle referred to the path of the particle. When at 1, 2, 3, 4, 5, 

 &c. in the orbit, the particle is also at 1, 2, 3, 4, 5, <Src. in the wave sur- 

 face. Thus the points which succeed each other towards the right on the 

 path, succeed towards the left on the wave form. 



^gs. 7 and 8 represent the genesis of the negative wave of the first or- 

 der. A solid Q 2 reposes suspended in the fluid, and is suddenly raised 

 out of it. A negative wave is generated and propagated along the chan- 

 nel, as W I in figs, 8, 9 and 10. This negative wave of the first order, 



2c2 



