528 Sir William Thomson on Sationary 



The motion represented by this solution, with any values of 

 and C, is steady and stable throughout any finite length of the 

 canal on each side of the ridge, provided the water is introduced 

 at one end of the portion considered and taken away at the 

 other conformably. If the canal extends to infinity in both 

 directions, and if the water throughout be given in the state of 

 motion corresponding to the solution (46) ; the motion through- 

 out any finite distance on each side of the ridge will continue 

 for an infinite time conformable to (46) . The water, if given at 

 rest, might be started into this state of motion in the follow- 

 ing manner : — First displace its surface to the shape repre- 

 sented by equation (46), and apply a rigid corrugated lid to 

 keep it exactly in this shape, so that it is now enclosed as it 

 were in a rectangular tube with one side corrugated, two 

 sides plane, and the fourth side (the bottom) plane, except at 

 the place of the ridge. Next by means of a piston set the 

 water gradually in motion in this tube. To begin with, the 

 pressure on the lid will, in virtue of gravity, be non-uniform ; 

 less at the high parts and greater at the low parts. If too 

 great a velocity be given to the water by the piston the pres- 

 sure will, in virtue of fluid motion, be greater at the high 

 parts and less at the low parts. If the average velocity be 

 made exactly U, the pressure will be uniform over the lid, 

 which may then be dissolved ; thus the liquid is left moving 

 steadily under the surface represented by equation (46) as 

 free surface. But it is only in virtue of this motion being 

 given to the fluid throughout an infinite length of the canal 

 on each side of the ridge, that the motion can remain steady 

 on each side of the ridge conformable to (46), except for the 

 particular case of this general solution, corresponding to 



C=0 and <y=,i£gg.» x N, . . . (47), 

 which reduces (46) to 



00 e.-x 



A/D cr-tX °° 



i = - — L- (o- 1 N 1 sin -~ + i 20N.e d) when x is positive 



1 + VJO JJ 2 



e-x 



and, fo= 2 '-, S^N-eD when a? is negative 



(48); 



this being the practical solution for the case of water flowing 

 from the side of x negative over the single ridge and towards 

 the side of x positive. It is the mathematical realization, for 

 the case of a single ridge, of the circumstances described in 

 Part I. No. 1 {ante, pp. 356-357), and is the mathematical 



