Waves in Flowing Water. 519 



of the present article, and will be more fully developed in 

 Part IV. 



To find the steady motion of water flowing in a rectangular 

 channel over a bottom with geometrically specified inequa- 

 lities, it is convenient, after the manner of Fourier, to first 

 solve the problem for the case in which the profile of the 

 bottom is a curve of sines deviating infinitesimally from a 

 horizontal plane. 



For convenience, take OX along the mean level of the 

 bottom, positive in the direction of U the mean velocity of 

 the stream ; and OY vertical, positive upwards. Let 



h=TL cos mx (1) 



be the equation of the bottom ; and 



y — J) = i) = ^cosmx (2) 



be the equation of the free surface, \) being height above its 

 mean level. Let cp be the velocity potential ; ?£, v the velocity 

 components ; and p the pressure at any point (x, y) of the 

 water at time t : so that we have 



«=gand,= g (3), 



and 



p=C-ffy-i(v? + T?) (4). 



Now the deviation from uniform horizontal velocity is infini- 

 tesimal, and therefore v and w — U, are infinitely small. Hence 

 (4) gives 



2> = C-gy-iIP + U(u-U) . . . . (5). 



<f) must be a solution of the equation of continuity -~ + —~ = 0, 



and the proper one for our present case clearly is 



<£ = U#+ sinm^Ke^ + K'e-"^) ... (6), 



where, because the motion is steady, K and K7 are constants. 

 This, in virtue of (3), gives 



u — U = mcoswtf (Ke^ + K'e-^), . . (7) ; 



v = msmmx{K6 m y-K'€- m y) ... (8). 



Hence, as the values of y at the bottom and at the surface are 

 infinitely nearly and D respectively, we find respectively 

 for the vertical component velocity at the bottom and at the 

 surface, 



m sin mx (K — K7) , and m sin mx (Ke mD — K f e~ mI> ) . 



Hence, to make the bottom-stream-lines and surface-stream- 



2N2 



