228 VARIABLE MOTION. 



123. Variable Motion. In every stream in which the 

 discharge is constant for a given time, the velocity at differ- 

 ent places depends on the slope of the bed. In general, the 

 velocity will be greater as the slope of the bed is greater; 

 and, as the velocity varies inversely as the transverse section 

 of the stream, the section will be least where the velocity 

 and slope are greatest. In a stream in which the velocity 

 is variable, the work due to the fall of the stream for a given 

 distance is equal to the work destroyed by friction together 

 with the kinetic energy corresponding to the change of 

 velocity, i. e., the whole fall is the sum of that expended in 

 overcoming friction, and of that expended in increasing the 

 velocity, when the velocity increases, or if the velocity de- 

 creases, the head is the difference of these quantities.* 



The resistance of friction upon a small portion of the 

 length of the stream may be regarded as constant and meas- 

 ured by a head of water 



= /*- (1) 



7 a 2g v ' 



Let ABCD represent a longitudi- 

 nal section of a short portion of a 

 stream, AB the surface of the stream, 

 and AE and HG two horizontal 

 lines. Let I = the length of AB in 

 feet; li = BE, the fall from A to 

 B ; v = the velocity of the stream at the upper section 

 AD ; and v^ = the velocity at the lower section BC. 



Now the velocity of any particle B, at the surface of the 

 stream, is due to the height h, together with the velocity at 

 A ; hence we have, for its velocity v 1 , 



~ 



* In long rivers, with slopes not greater than 3 feet per mfle, the velocity head 

 is usually insignificant compared with the friction head. (See Fanning's Water- 

 Supply Engineering, p. 803.) 



