147 



Equation (182) is the general equation of continuity in a tidal 

 cbannel. 



298. Since Q=vX=vzD, equation (182) may be written: 



(i{vzD)l(ix+z()ylbt=^ (183) 



If a channel is of both constant width and constant depth below 

 mean tide level, and the tidal fluctuation is so small with respect to 

 the depth that the variation in D may be neglected, equation (183) 

 becomes: 



zB^lbx+z^iylbt=0 

 or: 



Z>dy/da;+d?//d^=0 (184) 



299. Distinction between mean depth and hydraulic radius.- — Chan- 

 nels, whether natural or artificial, are usually so wide with respect to 

 their depth that, if the tide does not overflow the banks of the channel 

 proper, the mean depth, D, in the equation of continuity does not 

 differ materially from the hydraulic radius, r, in the friction term of 

 the equation of motion. On the other hand, if the channel is bordered 

 by tide flats and sloughs, in wliich water is stored and released as the 

 tide rises and falls, but which carry no appreciable current, the value 

 of D may be much less than the value of r. In other words, D is 

 computed from the gross width of the channel and r from the net 

 width after deducting areas which carry no substantial flow. 



CUBATURE OF A CLOSED CHANNEL 



300. Method of cubature. — The currents in a closed tidal channel are 

 caused by the filling and emptying of the tidal prism, and by the fresh- 

 water discharge from any rivers and streams which may enter it. By 

 taking simultaneous readings of the height of the tide at a sufficient 

 number of stations between a given tidal station and the head of tide, 

 the changes in the volume of water in the tidal prism from hour to 

 hour, or at shorter intervals, may be computed, and the positive and 

 negative discharges at the station due to the filling and emptying of 

 the tidal prism ascertained therefrom. The total discharge is then 

 the algebraic sum of the tidal discharge and the measured or estimated 

 upland discharge. The mean velocity at the station at any given 

 time may be deterniined by dividing the total discharge by the area 

 of the cross section at the station at that time. Tliis process is termed 

 the cubature of the channel. It is essentially the arithmetic integra- 

 tion of the general equation of continuity. 



301. Basic data. — The tidal stations established for a cubature 

 should be spaced at such distances that no material error is intro- 

 duced by taking the water surfaces between them as planes. This 



