318 



HYDRAULICS AND ITS APPLICATIONS 



Here ^ -, is positive .*. h increases down-stream. Down-stream 



j f jt^3 



. - tends to the limiting value unity, so that in this direction 



<l h . 



the limiting value of .is /', or the surface tends to become horizontal 



'/ / 



(Fig. 141). 



Up-stream, as the depth 



^ H, and, for this value of h, 



reach a point where 

 = x or the surface curve 



diminishes, we 



djt 



I - d I 



here becomes perpendicular to the bed of the stream, a standing wave 

 being produced. 



This is the curve obtained where an under-water obstruction such as a 

 dam or broad-crested weir is placed across a stream of rapid slope. 



Since the possibility of -r. becoming infinite, depends on - being 



greater than unity, the production of a standing wave under these 



circumstances is only possible 

 where this latter condition is satis- 

 fied. 



In practice the two most im- 

 portant cases are those represented 

 in 1 (b) and 1 (c). In the first of 

 : i these, the effect of a sudden drop 

 in the bed of a stream may, as 

 already explained, be serious, while 



the case of the reduction in level in a fore-bay feeding a power plant, 

 caused by the sudden demand for energy by the turbines also comes under 

 this heading. In the second, the effect of a dam in increasing the surface 

 elevation at points further up-stream is important. The investigation of 

 each of these cases resolves itself into determining, from a solution of 

 equation (6;, the value of li corresponding to any point at a distance / 

 from some datum, since when this is known, the rise or fall from normal, 

 and consequently the change in velocity, can be determined. 

 To obtain a solution for the equation, we have 



Via. 141. 



- 1 



dl 



7/ a 



2t 



7"J 



(6) 



