112 



at the given time. The second term, {v/g)bv/bx, is the rate of change 

 of v^/2g and may therefore be regarded as the component slope due to 

 the velocity head. The third term represents the effect of the accelera- 

 tion or deceleration of the current, and the last term is due to the 

 frictional resistance. 



The velocity, v, at a given instant, varies in fact from point to 

 point in a cross section of a channel carrying tidal flow, as it does 

 in a cross section of a channel when the flow is steady. In both cases, 

 V is taken as the mean velocity at the section. 



In the derivation of equation (112) the flow is regarded as contin- 

 uously turbulent, even during the short interval in which the velocity 

 becomes very small in passing through zero, as the current reverses. 

 It is evident, however, that a change in the character of the flow during 

 so brief a period may be disregarded, even if such change in fact 

 occurs. 



222. Application of general equation to steady uniform flow. — When 

 the flow is steady and uniform, the velocity throughout the channel 

 remains constant, and bv/dx and bv/bt are zero. Taking the velocity 

 as in the positive direction, equation (112) reduces to 



by/bx+v'IC'r^O. (113) 



Designating the slope of the water surface as s, and observing that 

 when the flow is steady the slope is downward, so that by/bx=—Sy 

 equation (113) becomes: 



Whence _ 



v=C^/Fs. (114) 



Equation (114) is the generally accepted basic formula for steady 

 flow, in which the Chezy coefficient, C, may be determined from the 

 Kutter, Bazin, Manning, or other formulas. 



223. Selection of Chezy coefficient for tidal flow.- — It is apparent from 

 the preceding discussion that the value of C to be used in equation 

 (112) when the flow is tidal should be that applicable to the channel 

 were the flow steady. While the value of C determined from the 

 Kutter formula varies somewhat with the slope in the channel, and this 

 slope fluctuates between limits when the flow is tidal, this variation in 

 C is so small with the slopes usually found in tidal channels that either 

 the maximum or the numerical mean or median slope during the tidal 

 cycle may be used in the application of the formula without affecting 

 the value of O to a greater degree than that inherent in the uncertainty 

 in the selection of the proper coefficient of roughness. 



224. Expression for the friction term lohen the velocity has a harmonic 

 fluctuation. — It has been seen that the friction term in equation (112) 

 changes its sign in passing through zero, while the expression for the 

 friction term, v^jC^r, does not change its sign. A mathematically con- 



