Waves in Flowing Water. 357 



give rise to stationary corrugations large or small, but per- 

 fectly stationary, however large, short of the limit that would 

 produce infinite convex curvature (according to Stokes's theory 

 an obtuse angle of 120°) at any transverse line of the water 

 surface. 



2. But in particular cases the water flowing away from the 

 inequalities may be perfectly smooth and horizontal. This is 

 obvious because of the following reasons : — 



(i.) If water is flowing over plane bottom with infinitesimal 

 corrugations, an inequality which could produce such corru- 

 gations may be placed on the bottom so as either to double 

 those previously existing corrugations of the surface or to 

 annul them. 



(ii.) The wave-length (that is to say the length from crest 

 to crest) is a determinate function of the mean depth of the 

 water and of the height of the corrugations above it, and of the 

 volume of water flowing per unit of time. This function is 

 determined graphically in Stokes's theory of finite waves. It 

 is independent of the height, and is given by the well-known 

 formula when the height is infinitesimal. 



(iii.) From No. ii. it follows that, as it is always possible to 

 diminish the height of the corrugations by properly adjusted 

 obstacles in the bottom, it is always possible to annul them. 



3. The fundamental principle in this mode of considering the 

 subject is that whatever disturbance there may be in a perpet- 

 ually sustained stream, the motion becomes ultimately steady, 

 all agitations being carried away down stream, because the 

 velocity of propagation, relatively to the water, of waves of 

 less than the critical length, is less than the velocity of flow of 

 the water relatively to the canal. 



In Part II., to be published in the November number of 

 the Magazine, the integral horizontal component of fluid 

 pressure on any number of inequalities in the bottom, or bars, 

 will be found from consideration of the work done in genera- 

 ting stationary waves, and the obvious application to the work 

 done by wave-making in towing a boat through a canal will 

 be considered. The definitive investigation of the wave- 

 making effect when the inequalities in the bottom are geo- 

 metrically defined, to which I have just now referred, will 

 follow ; and 1 hope to include in Part II., or at all events in 

 Part III. to be published in December, a complete investiga- 

 tion, illustrated by drawings, of the beautiful pattern of waves 

 produced by a ship propelled uniformly through calm deep 

 water. 



