229 



The integration of which gives, since f=0 when x=0: 



^=(SIA)x. (300) 



It will be convenient to place: 



S/A=n. (301) 



So that: 



t=nx. (302) 



The equation of the current at any cross section of the channel may 

 be written: 



v=B sin (a^+jS) 

 in which, from equation (150) : 



^ = H°-ct>-'ir/2 

 From equations (299) and (302) : 



^=—nx—<l) — Tr. 



The equations of the tide and current in an ideal estuary are then: 



y=A cos iat—7u) (303) 



v=B sin {at — nx— 0— x) = — 5 sin (at — nx—cp). (304) 



These equations show that the tides and currents progress up an 

 ideal estuary at the constant rate of a/n. 



440. The differential equation of the primary current has been 

 derived in equation (142), paragraph 243: 



by/()x+(l/g)bv/bt^i8/3Tr)Bv/C'r=0 



Substituting the differential coefficients and the expression for v 

 obtained from equations (303) and (304): 



An sin {at—nx) — {aBjg) cos (at — nx—cf)) 



-(S/St) (ByC'r) sin {at—nx-cl>)=0 (305) 



By placing at—nx=0, the equation of condition is derived: 



-(aB/g) cos </)+(8/37r) {B-/C'r) sin = (306) 



and by placing at—nx=Tr/2: 



An- (aB/g) sin 0-(8/37r) (B'/C'r) cos 0=0 (307) 



