CHANGES IN CROSS SECTION 



121 



After dividing through by D2 and rearranging terms, we obtain 



F2V2g 



=f-:(-^)-' 



But since Q is constant, 



ViDiWi = V2D2W2 



and 



Pi ^ 1^2 F2 ^ my^ 

 P2 " W^iFi ~ Wi/ 



/V2V2gV" 



D, 



Vi'/2g 



\ D, 



Substituting in equation (1009), 



V2V2g 

 Do 





D, 



ViV2g 



{l^^)-l [1010] 



Solution of equation (1010) for numerical computation is awkward. 

 The authors have plotted the equation in Fig. 1003, which shows the 



relationship between 



ViV2g V2V2g 



, and W2/W1 over the range of 



Di ' D2 



practical values of the variables. Examination of Fig. 1003 shows that 



FiV2g . V2^/2g 



for every value of — ;;^ — there are two possible values of 



cept when 



V2' 



Di 



D, 



ex- 



/2g 



D, 



is equal to one-half — that is, when the flow in the 



downstream section is critical. This is to be expected, for unless the 

 flow in the downstream section is critical, two alternate depths of flow 

 are possible. The diagram also shows that for any given ratio of velocity 

 head to depth in the downstream section, there are two possible values 

 of the ratio for the upstream section, unless the flow in the upstream 

 section is critical. 



Since there is no loss of energy, flow could be in either direction, and 



the diagram should be symmetrical about the line 



FiV2g V2V2g 



Di D2 



with the values of the W2/W1 curves on one side of the line equal to the 

 exact reciprocals of the curves symmetrically opposite, on the other side 

 of the line. The curves shown in the diagram are not exactly sym- 

 metrical, however, for the values of W2/W1 plotted, above and below 

 unity, are not reciprocals of each other. 



