232 HYDRAULICS AND ITS APPLICATIONS 



When the retardation is gradual, part of the kinetic energy is absorbed 

 in doing work against the retarding force, and part in compressing the 

 column, the latter factor becoming increasingly important as the rate of 

 retardation is increased. 



(b) Sudden Stoppage of Motion Ideal Case. 



If a column of water, flowing with velocity v along a uniform pipe 

 (supposed rigid), have its motion checked by the instantaneous closure of 

 a rigid valve, the phenomena experienced are due entirely to the elasticity 

 of the column, and are analogous to those obtaining in the case of the 

 longitudinal impact of an elastic bar against a rigid wall. 



At the instant of impact, the motion of the layer in contact with the 

 valve is suddenly stopped, and its kinetic energy is changed into 

 resilience, or energy of strain, with a consequent sudden rise in pressure. 

 This stoppage and rise in pressure is almost instantaneously transmitted 

 to the adjacent layer, and so on, the state of zero velocity and maximum 

 pressure (this at any point being p' above the pressure obtaining at that 

 point with steady flow at velocity v) being propagated as a pressure- wave 

 along the pipe, with velocity V p . \_V P is the same as the velocity of 

 sound through water, i.e., about 4,700 ft. per second, depending slightly 

 on temperature.] 



This wave reaches the open end of the pipe after t seconds, where 

 t = I -r- V p . At this instant the whole of the column is instantaneously 

 at rest in a state of compression. 



At the open end, however, a constant pressure^ is maintained, and in 

 consequence the strain energy of the end layer is reconverted into kinetic 

 energy, this (neglecting losses), rebounding with its original velocity v 

 and with the normal pressure obtaining at this point under a state of 

 steady flow towards the open end with this velocity. 



This state of normal pressure and of velocity ( v) is then propagated 

 as a wave towards the valve, reaching the latter after a second interval 

 I -f- V p seconds. At this latter instant the whole of the column is 

 unstrained and is moving towards the open end with velocity v. At the 

 same instant the motion of the layer nearest the valve is stopped, and a 

 wave of zero velocity and of pressure (p' below the pressure obtaining at 

 the point at the instant before the stoppage of the motion, or p below the 

 pressure at the point with no flow through the pipe) is transmitted along 

 the pipe to be reflected from the open end as a wave of normal pressure 

 and velocity v towards the valve. When this wave reaches the valve, 



