﻿Problem of the Weir. 781 



In the case where the lower stream-surface is depressed 

 (U 2 > t ^A)it maybe shown that waves of small amplitude 

 cannot be maintained in the lower stream. For in this case 





«r = l/ 



v ■< 



1 and W/gh > 1 



Hence, 



U 2 , 



U' 2 



. h'/li > g . A'/U' 2 , 



i. e., 







vt/v 2 >gh'/U' 2 , 



i. e., 







^>gh f /X}' 2 . 



Therefore, 







l^ffh'fll'*, 



or 







W 2 >gli. 



The velocity of propagation of irrotational waves of small 

 amplitude in a channel of depth It' is V, where 



Y 2 =gh'. tanh kJi'/kJl (k — 2ir/ wave-length). 

 Hence, V 2 always <r gh' 



i.e., V „ <U'. 



Thus, if such waves were ever formed they would be 

 carried away by the preponderating velocity of the lower 

 stream. 



§ 6. When U 2 -< gh the surface-level of the stream on the 

 lower bed is elevated, but it can be shown that this elevation 

 cannot exceed a certain limit, viz., half the depth of the 

 stream on the upper bed. 



For, in this case ot > 1 and the root chosen lies between 

 8 and 8 + 1 A. 



Hence, 8 < tn ■<: 8 + 1/k, 



or 1 + d/h < h'/h < 1 + d/h + U 2 /2gh. 



Also, in the present case, IP < gh. 



.'. h + d< //< k + d + h/2, 

 or 0< K' — (h + d) < h/2. 



Hence, the difference of levels of the upper and lower 

 streams is always less than half the depth of the stream on 

 the upper bed. 



§ 7. In the special case where /c = 8, i. e., 2gli\XJ 2 = 1 + d. h, 

 the cubic (12) admits of a very simple solution. For. in 

 this case, (12) becomes 



k^-<ct 2 {1 + k 2 ) + 1 = 0, 



or (/<>&■ — 1)(V J — /cot— L) = 0. 



