HYDRAULIC JUMP IN A CHANNEL WITH SLOPING BOTTOM 35 



A longitudinal section through a hydraulic jump on a sloping floor is 

 shown in Fig. 307. The side walls of the flume in which the jump 

 forms are assumed to be vertical and parallel. The unbalanced force on 



Fig. 307. Diagrammatic Longitudinal Section through a Hydraulic 

 Jump on a Sloping Floor. 



the water between sections 1-1 and 2-2 acts horizontally toward the 

 left, and is approximately equal to 



w{Hj-\- Di-\- L tan af wDi^ cos^ a P"'^ ^^ , . 

 I wDdA sm a 



-r 



where dA is an element of the area of the bottom, at depth D. The 

 time rate of change of momentum is 



wQ 



r^^°'"-/f, + Pi+Ltana) 



assuming that the velocity at section 2-2 is uniform and approximately 

 horizontal in direction. These two quantities could be equated, and the 

 resulting equation solved for Hj. It is unnecessary to do so, however, 

 for it is seen that Hj depends not only upon D, Q, and a, but also upon 

 the shape of the profile and the length of the jump. The latter factors, 

 which are unknown even for the jump on a level floor, and which must 

 certainly vary with a, exert a large influence upon the value of Hj. 

 Attempts to avoid this difficulty by considering the components of forces 

 and momenta along the slope are equally fruitless, for then the compo- 

 nent of the weight of the water between the initial and final sections must 

 be included. Without knowing the surface profile through the jump 

 this factor cannot be estimated. 



It is somewhat disappointing not to be able to obtain a formula as 

 simple as the well-verified formula for the height of the jump in a level 

 channel, but it is questionable whether such a formula would be of much 

 use if it could be derived. We know by experience that when the jump 



