﻿780 Mr. J. H. C. Searle on the 



Since r\ 5; according as /c ^ 2, 

 therefore U 7 ^ U ,, „ U 2 ^ ^because «r = 7i'/7i = U/U\ 



Also, U' 2 - U 2 = 2#( A + af— A') ; 



therefore li ^h + d according as U 2 ^ gli. 



Hence, the surface-level of the stream along the lower 

 bed will be higher or lower than the surface-level of the 

 upper stream according as U 2 > gh. 



It will be noticed that the " critical " velocity, ~[j 2 =g]i } is 

 that of a long wave in a canal of depth h, and it seems pro- 

 bable that when U has this value, or very nearly this value, 

 the motion will become unsteady on account of the tendency 

 of the stream-surface to adjust itself alternately to the higher 

 or lower level, according as U is slightly less than or 

 slightly greater than ((/A)*- Some such effect seems to have 

 been noticed by Bidone in his researches on the flow of 

 water over weirs. From a practical point of view the result 

 may be of use, because it shows that in this type of motion 

 the " critical " velocity should be avoided in order to prevent 

 "■scouring" of the channel. 



Lord Kelvin *, using a different method, arrived at the 

 approximate formula 



y=rf/(?D/V»-l); 



where y — elevation of surface ( = 7i 4- d— h' in present notation), 

 d= ,, stream-bed (small), 



D = a depth lying between h and li , 

 V — a velocity lying between U and U'. 



It appears that there is an elevation or depression of the 

 lower stream-surface according as V 2 -^ gD when d>0 and 

 inversely when d<0, i. e., when the stream-bed falls. 



In Lord Kelvin's work the values of V and D are neces- 

 sarily undetermined, although, when the irregularities of the 

 stream-bed are small, we may put approximately the values 

 h and U of the upper stream. The present note shows that 

 the elevation or depression of the lower stream-surface 

 depends accurately on the criterion U 2 ^ gli, and also does 

 not depend on d being small. 



If we reverse the general direction of the stream the 

 theory indicates a depression of the surface when a stream 

 flows from a lower to a higher bed, — a phenomenon often 

 observed m practice f. 



* L. c. ante. 



t Merrimazi, Hydraulics, p. 295. 



