236 HYDRAULICS AND ITS APPLICATIONS 



and w m , V and V m , a and a m be the weights of unit volume of, th< 

 velocities of wave propagation in, and the sectional areas of the wate: 

 column and metal of the pipe wall respectively. 



Then, with instantaneous closure the ends of the water and meta 

 columns move, at impact, with a common velocity u, and waves 

 respectively of compression and of extension, travel along the wate 

 column and the pipe wall. 



Hence, after a very short interval of time 8 1, lengths V B t and V m 8 t o 

 the water column and of the pipe will be moving with velocity u, and th< 

 equation of momentum gives : 



[w a V + w m a m V m } u B t = w a V r S t 



( 



. . u = v 



w m a 



Each element of the column and of the pipe, as the wave passes it, takes 

 suddenly the velocity u, while each element of the water column takes th< 



compression p and therefore the stress (v u) \J , and eacl 



element of the pipe takes the extension -p- and the stress u \/^ 



' -m ' 



Substituting for u we have the pressure rise in the water given by 



\/ w K' ibs. per square foot. 



this may be written 



r= - 1 Ibs. per squar 



a . / _L . . 



a V E w 1 



Ibs. per square foot, 



(2; 

 A" w 



1 Imagine a bar of unit cross-sectional area to impinge with velocity v in the direction oj 



', 8 t , 



its axis, against a rigid wall. After a very short interval 5 t seconds, a mass - 



been brought to rest, and, if p is the (uniform) pressure on the end of the bar during this 

 interval we have, equating the force x time, to the change of momentum : 



p S t = Wm V S * . v <ap -" .T 



'' 



Butp = r \/ L^i 1 , so that, equating these two expressions we get V m . = \J '' 

 per second. 



