190 



The equations of the entrance tides are: 



?/o=A cos (at+ao) yi=Ai cos {at^ai). 



If the instantaneous profiles were straight hues the equation of the 

 tide at a point in the canal distant x from the initial end would be: 



^ cos (at^a)=yo-\-(,x/L) (7/1— 7/0)= A cos (at + ao) 



+ {ilL)[A, cos {at+ao)-A, cos (at^ao)]. (272) 



The coordinate amplitudes of the tide at the point x are then: 



A sin a=Ao sin ao-{-{xlL) (Ai sin ai — Ao sin uq) (273) 



A cos a=Ao cos ao-{-(x/L) {Ai cos a^ — Ao cos ao). (274) 



367. First computation of primary currents in subsections. — The 

 primary current in each subsection may be taken as that at the middle 

 of the subsection, and the midpoints of the successive subsections will 

 be designated the velocity stations. The amplitude, H, and the initial 

 phase, H°, of the head, and the amplitude, S, of the slope in the middle 

 subsection of the canal are computed, as described in paragraph 239, 

 from the components of the tide at the ends of the subsection, derived 

 from equations (273) and (274). The amplitude, Bq, the initial phase, 

 jSo, and the resulting coordinate amplitudes, Bq sin jSq a,nd Bq cos (80, of 

 the primary current at the velocity station at the middle of the sub- 

 section are then computed by the process set forth in paragraphs 246 

 and 248. 



368. The corresponding primary currents at the other A^elocity sta- 

 tions are determined by the general equation of continuity (equation 

 182): 



c)Qlc)x^zc)y/dt=0. 



Since differential equations remain approximately true when small 

 finite increments are substituted for the differentials, equation (182) 

 may be written 



AQlAx+2by/dt = 

 or 



AQ=-zAxc)y/()t. (275) 



In this equation AQ is the algebraic increase, at any instant,* in the 

 discharge from one velocity station to the next, z the mean width of 

 water surface between the stations at mean tide and A a:; the distance 

 between the sections. It should be noted that Ax may be large when 

 expressed in feet while being small in relation to the change which it 

 produces in the discharge. 



