WATER HAMMER 281 



of efflux at the latter instant. This only holds so long as the valve is 

 opening. Suppose the valve to be stopped at an instant when its opening 

 is a</, the velocity of efflux at this instant being calculated to be r</. 

 Let a ' -7- (i = n. 



From this time onwards -7-r 2 = n . -r-A and equation (9') becomes 



d- t d t 



d V 



So that 



2qh 2ln 



where c = : 77- ; k = ^ : 



m 

 Integrating we get 



1 



When i = 0, i.e., immediately the valve comes to rest, VQ = 

 Using this to determine D, we finally get 

 V7 + VQ ' 2 -^Z t 



/ feet P er second 



V7--V* 



as the velocity of efflux after t seconds from the stoppage of the valve. 

 It will be noted that as t increases, this tends to the limit 



\/7 = 



The method of treatment so far outlined, while giving results which 

 are rigorously true for an incompressible fluid in a rigid pipe line fails 

 to account for many of the phenomena actually observed during the 

 stoppage of motion in a long column of water, since these are largely due 

 to the elasticity of the water column. For example, an examination of 

 equations (1) or (2) indicates that an instantaneous stoppage of motion, 

 involving an infinite retardation, will necessitate an infinite retarding 

 force, and hence an infinite pressure at the closed end of the pipe, a 

 conclusion which is not at all borne out by the result of experiment. 



Actually, when the column is brought instantaneously to rest, 

 compression takes place ; a wave of compression is reflected from the 

 closed end of the pipe ; and the initial kinetic energy of the water is 

 transformed into resilient energy or energy of strain. 



