127 



is negative, or vice versa; and is running uphill when both the head 

 and the velocity have the same algebraic sign. While this convention 

 may appear unnatural, it removes the confusion that would result 

 were vertical distances taken as positive in a downward direction. 

 In each of the diagrams illustrating the preceding examples, the head 

 reaches a maximum at the time marked C, and the primary current 

 reaches its strength, in the opposite direction, at a subsequent time 

 marked D. The strength of the primary current in a tidal channel 

 therefore lags behind the maximum surface head and slope by the 

 time interval CD. The turn of the primary current lags behind the 

 turn of the surface head and slope by the equal interval AB. 



Designating the time C as ^o, and the time D as U, then from the 

 equation of the surface head (equation 136): 



hs^H cos {at+H") 



it is evident that 



From the equation of the primary current (equation 163): 



v^Bsin (at -^W-cj) -90°). 

 and 



ati-\-H^-(t>-90°=-90°. 

 Whence 



ati — aio=0- 



The intervals CD and AB are then equal to <^/a ; and the angle <f) may 

 be designated the angular lag of the primary current. 



255. Characteristics of tidal flow. — In each of the preceding examples 

 the water flows downhill during the intervals indicated as BA on the 

 diagrams. At the moments marked A the water surface is level, but 

 the momentum of the moving water continues to carry it in the direc- 

 tion of its motion. During the intervals from A to B the water flows 

 uphill until the momentum is checked. At the instants marked G, 

 when the current reaches its maximum velocity in either direction, 

 the acceleration is zero, and the velocity is determined by the slope 

 and frictional resistance only, and is the same as though the flow 

 were steady; but as the velocity lags behind the head, the maximum 

 velocity does not occur when the head is a maximum. When the lag 

 is very large, the maximum velocity occurs at a moment when the 

 head is very small. 



