no • 



218. Derivation of equation of motion. — Taking the X axis of coordi- 

 nates as a horizontal line in the direction of the axis of the channel, and 

 the Y axis as vertical, let: 



V be the velocity of the current at the time t, and at a cross 

 section of the channel distant x from the origin of coordinates ; 

 V is taken as positive when the direction of flow is in the 

 positive direction of x, and negative when the flow is in the 

 opposite direction. 



dv/dt, the acceleration of the velocity at a given cross section 

 with respect to time. 



br/dx, the rate at which the velocity is increasing (algebraically) 

 at a given instant with the distance of the cross section from 

 the origin. 



dx, the distance, along the direction of the X axis, traveled by 

 a particle of water during the elementary time interval dt. 



{dyldx)dx, the (algebraic) increase in the elevation of the water 

 surface in the distance dx; this increase being positive if the 

 slope of the water surface is upward, and negative if down- 

 ward in the direction x positive. 



X, the area of the cross section of water prism of the channel, 

 at the point x, and at the time t. 



Q, the discharge through the cross section. 



w, the weight of 1 cubic foot of water. 



g, the acceleration due to gravity. 



m, the mass of the water discharged through the cross section 

 during the time interval dt. 



r, the hydraulic radius of the channel at the section under 

 consideration. 



C, the Chezy coefficient applicable to this section. 

 Then: v=dx/dt 



Q=Xv=Xdx/dt. 



The volume of the discharge, during the time dt is Qdt = Xdx, and 

 its mass, m, is: 



m = wQdt/g = wXdx/g. 



The mass of the water in an elementary section of the channel of 

 length dx is also : 



wXdxlg=m. 



219. During the time interval dt, work is done in an elementary 

 section of the channel of length dx: 



(a) In raising the mass of the discharge the distance {dy/c>x)dx in its 

 passage through the channel. 

 The work so done is : 



mg{c>ylc)x)dx. 



