171 



Obviously, under such conditions, the total surface head through 

 the canal is always zero, and the curients are due solely to the filling 

 and emptying of the canal from the two entrances. At the middle of 

 the canal the currents disappear, and the tides are said to meet. 



FRICTIONLESS TIDES AND CURRENTS IN A CLOSED CANAL 



335. The amplitude and initial phase of each component of the tide 

 at the head of a closed canal is determined by the condition that the 

 currents are there zero. Taking the open end of the canal as the initial 

 entrance, the equation of a component of the tide at this entrance is 

 yo=Ao cos (ai + ao) and at the other end, at the head of the canal, yi = 

 Ai cos {at-\-ai). At the head of the canal x=L, the length of the canal, 

 and the velocity of the corresponding component of the current is, 

 from equation (230) : 



v=(g/c)Ao sin (ai + Q:o)/sin ■y—{g/c)Ai sin (ai + ai) cos 7/sin 7 = 



Whence : 



Ai sin {at-{-ai)=Ao sin (ai+o:o)/cos 7 (248) 



Since this equation is identically true for all values of t: 



A,=Ao/ cos 7 (249) 



ai = ao (250) 



Substituting these equivalents in equation (228), the equation of a 

 component of the tide at any point in a closed canal becomes: 



y=Ao cos (a^+ao) sin( 1—j^/i) 7/sin 7 



+ ^4o cos {at->rao) sin (x/Z)7/sin7 cos 7 

 —Aq cos (at-^ao) [(sin 7 cos (x/L) 7 — cos 7 sin {x/L)y) 



H-sin {x I L)'y]/ sin 7 cos 7 

 =Ao cos (a^+Q:o)[sin 7 cos {xlL)y cos 7 



+ (1 — cos^ 7) sin {xlL)y]/sm. 7 cos 7 

 — Ao cos (ai +0:0) [cos {x/L)y cos 7 + sin (x/L)y sin 7] /cos 7 

 =^0 cos (at-\-ao) cos (1— x/Z)7/cos 7. (251) 



The substitution of the same values of Ai and ai in equation (230) 

 gives the equation of the corresponding component of the current, 

 which similarly reduces to : 



v= — {g/c)Ao sin (at+ao) sin (1— a'/Z,)7/cos 7 (252) 



336. The form of equations (251) and (252) shows that, if the flow 

 were frictionless, high water and low water each would occur simul- 

 taneously throughout a closed canal, and the current would turn at 

 these instants at every point in the canal. The maximum currents 



192750—40 12 



COS 7 



