96 Formula for the Discharge over a Broad-crested Weir. 

 equating -j- to zero. This gives £=§H, and on substituting 



this value we get Q = -3850 sj2g . H* c.f.s. Writing this 



in the usual form, Q = | Gb \ / 2gH> = K6H* we have 

 C =-578 and K = 3'087. 



This method of treatment becomes more rational if account 

 be taken of the fact that in a parallel stream flowing in an 

 open channel, the distribution of velocity over any vertical 

 is not uniform, being a maximum at or near the surface and 

 a minimum at the bottom. 



Experiments show that the ratio of the mean velocity over 

 the section of such a stream, to the maximum surface velocity, 

 while varying with the depth, width, and roughness of the 

 bottom of the channel, lies between the limits 'S2 and '87 

 for such surfaces and depths as are common on the 

 crests of such weirs, this ratio increasing with the depth 

 of water. 



Assuming, as is practically the case, that the maximum 

 surface velocity in the case of the weir is equal to \ / 2g(H — t), 

 the mean velocity will equal k \/2g(}I — t), and the discharge 

 will be given by 



k . K . bH.% = 3-087*fflt = K'bH* cub. ft. sec. 



Thus corresponding to the values '82 and *87 of k, the values 

 of K' become 2'53 and 2*69. 



The validity of this formula receives remarkable con- 

 firmation from the results of a large series of tests on such 

 weirs, carried out in 1903 at Cornell University for the 

 U. S. Geological Survey *. From a summary of these tests 

 it appears that on broad weirs, for depths between 1 and 

 5 feet, the coefficient K' is sensibly uniform, increasing 

 slightly with H. With weirs from 5 to 16 feet wide K / lies 

 between 2*62 and 2*64, while with heads between *5 foot and 

 TO foot, K' varies from 2 73 to 2*64. Experiments at 

 Cornell University in 1899, on a weir having a crest 

 6*56 feet wide, with a sharp upstream edge f , show the 

 following results : — 



Value of H 



•5 



10 



1-5 



2-0 



5-0 



Value of K' 



2-49 



2-59 



253 



247 



2-69 



* Water Supply and Irrigation Paper, No. 200. U. ■ S. Geologic? 

 Survey. 



f Trans. American Soc. Civil Engineers, 1900. 



