121 



as corrections to a harmonically varying primary current, represented 

 by equation (115) 



v=B sin (a#+/3) 



243. Dropping the velocity head term, {vlg)'d v/dx, from equation 

 (112), and substituting for the friction term its principal component 

 for an harmonically varying current (equation 125), the differential 

 equation of the primary current is: 



^y/dx+ (l/g) dv/dt-{- {8/3T)Bv/C'r=0 (142) 



In this equation C and r are the values of the Chezy coefficient 

 and hydraulic radius at mean tide. 

 In equation (142) : 



c)y/bx=hs/l={H/l) cos {at +H')=S cos {at+H'), (143) 



in which S=H/l is the numerical value of the maximum slope in the 

 channel during the tidal cycle. 

 From equation (115): 



dv I dt =aB cos (at -{-^) (144) 



Equation (142) then becomes: 



Scos iat-hH') + (aB/g) cos (ai+/3) + (8/37r)(57CV) sin (a^+/3)=0 



(145) 

 In which a is expressed in radians per second. 



244. Solution of equation. — By placing at=0 and a^= — 7r/2 in 

 equation (145), two equations of condition are established from 

 which expressions for B and /3 may be derived. When at=0, equation 

 (145) reduces to: 



S cos H°+ (aB/g) cos ^+ (8/37r) (ByC'r) sin ,3=0 (146) 



and when af= — 7r/2, to: 



S sin iJo+ {aB/g) sin /3- (8/37r) (B'/C'r) cos ^3 = (147) 



Multiplying equation (146) by cos (3 and equation (147) by sin /? 

 and adding: 



^^(cos HO COS iS+sin H° sin ^) + (aB/g) {cos' /S + sin^ (3)=0 



or: 



Scos {W-^)+Ba/g=0 (148) 



Multiplying equation (146) by sin j8 and equation (147) by cos (3 and 

 subtracting: 



