314 HYDRAULICS AND ITS APPLICATIONS 



If h-2 hi = jo this reduces to 



<7 



A / i 2 . 2 hi ?'i 2 3 , 

 .'. *==V T + -2*1. 



This gives the height of the standing wave. 



An explanation of the production of the standing wave may be found 



as follows. An examination of equation (3) shows that -^-j can only be 

 infinite and a standing wave formed when f 2 = g h, or when 



^h dl = dTl' 

 But 



t; 2 d h v v d h _ r d v _ <1 ( v* 



g h' d I g ' h ' d I ~ // d I ~ d I \'l g 

 so that the standing wave is produced when 



__ 

 " dl\%g) ~ d I 



i.e., when the rate of decrease of kinetic energy is equal to the rate of 

 increase of potential and pressure energy due to an increase in the depth, 

 or rice versa. 



Until this point is reached the rate of decrease of kinetic energy is 

 greater than that of increase of pressure and potential energy, the 

 difference being due to energy expended in eddy formation. Assuming 

 for the moment that the surface curve could be continued through the 

 point, we should have the rate of increase of potential and pressure 

 energy greater than that of decrease of kinetic energy, and hence should 

 have an actual increase in total energy, a state of affairs which is 

 manifestly impossible. 



This can only be overcome by a sudden change in the distribution 

 of pressure over the section of the stream, the effect being almost identical 

 with that produced by the introduction of a solid obstacle in the path of 

 the stream. As a consequence of the shock thus produced there is a 

 sudden loss of energy in eddy production, the velocity of flow of necessity 

 falls, and a corresponding rise of surface ensues. 



> \/^ // 



After rising to the level, // 2 , where //.> ^ f , we have the state 



<H 

 of affairs considered in Case 1 (b). 



