26 HYDRAULIC JUMP IN RECTANGULAR CHANNELS 



from which may be obtained 



r>2' + D^D^ = ?^ [302] 



or, after substituting ViDi for Q, 



^2= -y±J^^^^:^^+^ [303] 



.. = -^ 



1^ fe'_u^ 



By substituting Q/Di for Fi in equation (302) we get 



■ 



0,D.(^l±^^9l^I,. [304] 



2 g 



Equation (304) shows that if Di = Dc, I>2 also equals Dc. The 

 equation is symmetrical in Di and D2. If Di is less than Dc, D2 must 

 be correspondingly greater than Dc, and vice versa. There seems to be 

 no physical phenomenon corresponding to a reversal of the jump in 

 direction. Therefore, there can be no jump unless Di is less than Dc, 

 and the jump, when it occurs, always takes place across the critical 

 depth.^ As already illustrated in Fig. 202, water can flow at depths 

 less than the critical depth without necessarily forming a jump. This 

 will be more fully discussed later in connection with backwater curves, 

 where the conditions determining the formation of a jump will be more 

 completely stated. 



The equation of the hydraulic jump can be written in a great many 

 different ways, in terms of the different possible fundamental and 

 derived variables. If any two of the four simplest variables, Di, D2, 

 Vi, and V2, are given, the values of the others can be determined. 

 Figure 502, page 57, permits a direct graphical solution. 



The above demonstration rests upon a law of mechanics as well 

 established as any law of nature, as well proved, for example, as the law 

 of gravitation, and there can be no question of the validity of the results. 

 In the hydraulic jump there is.continuous violent impact, and by means 

 of the resulting turbulence a part of the kinetic energy is converted into 

 heat. 



For given values of Di and Dc, the value of D2 given by equation (304) 

 will always be less than that given by equation (207). This is because 

 the change in depths represented by equation (207) occurs without loss 



^ This statement applies only to the stationary jump. 



