129 

 For wholly frictionless flow, = 90°, and 



B=gS/a=gH/al (164) 



The amplitude of the fluctuations of the current then varies from 

 day to day directly as the head, and hence nearly as the amplitude 

 of the tide. 



Obviously, in the fiords of Alaska, where depths of 1,000 feet are 

 common, and in other deep channels such as the Florida straits, the 

 tidal flow is essentially frictionless. 



259. If the depths in a channel are small enough and the currents 

 sufficiently marked to be of consequence to shipping, the tidal flow is 

 not frictionless and the currents depend upon both the friction head 

 and the acceleration head, as indicated in the second example (fig. 40). 

 The so-called hydraulic state of flow is one of degree only, and merges 

 without distmction into conditions of flow in which the acceleration 

 head becomes of increasing importance. The maximum velocity, or 

 the "strength of the current" is always less than that which would be 

 produced by the maximum head were the flow steady. The accelera- 

 tion head acts as a brake on the currents as the friction head diminishes. 



DISTORTIONS OF PRIMARY CURRENT 



260. The primary current has been derived by taking the surface 

 slope as a simple harmonic fluctuation; dropping the velocity head 

 term from the general equation of motion (equation 112); substituting 

 for the friction term its principal harmonic component (8/3 x) BvjC^r; 

 and taking the hydraulic radius, r, and the Chezy coefficient, C, at 

 mean tide. The corrections for these approximations will now be 

 developed. These corrections produce a velocity-time curve which is 

 more or less distorted from the simple harmonic curve of the primary 

 current. 



261. Corrections jor the variation of frictional resistance with the re- 

 versing square of the velocity. — The corrections to fulfill the condition 

 that the friction term is ±iV'^/C^r, may be computed, to any desired 

 degree of refinement, by a somewhat laborious process explained in 

 detail in appendix II. As there shown, these corrections, designated 

 as i, are proportional to the amplitude, B, of the primary current and 

 depend upon its angular lag, 4>, and its phase, at-\-^. The correction 

 factors, i/B, as so computed for successive values of 0, and for values 

 of a^+jS from to 180°, are shown in table X. For values of o^+/3 

 between 180° and 360° the table is entered with a«+/3— 180° and the 

 algebraic sign of correction reversed. As will be seen from the table, 

 the corrections are small when <^ is large, and the flow consequently 

 is nearly frictionless. They become zero when 0=90°. 



