191 



Designating the area, z/^x, of the water surface, at mean tide^ 

 between successive velocity stations as U, equation (275) becomes: 



AQ=-Uby/dt. (276) 



369. Strictly speaking, the rate of rise of water surface, by/dt, should 

 be computed at the center of gravity of the water surface between the 

 two velocity stations, but its location need not be determined with 

 mathematical precision. If the canal has a constant width at mean 

 tide, the storage stations, at which by/dt is to be computed, are mid- 

 way between the velocity stations; and if the subsections are also all 

 of the same length, these storage stations are at the ends of the sub- 

 sections. If the width of the canal is tolerably constant, the storage 

 stations also may be taken at points half way between the velocity 

 stations; otherwise the location of the center of gravity of the water 

 surface may be roughly estimated and the storage stations selected 

 accordingly. 



The equation of the primary tide at a storage station is in the form : 



y=A cos {at-\-a). 

 Differentiating : 



by/dt=—aA sin (at-\-a). 



Substituting this value in equation (276) : 



AQ=aUA sin {at + a). (277) 



370. Designating respectively the area of the cross section of the 

 velocity station at the middle of the canal, at mean tide, as Mq; the area 

 at any other velocity station as M; the discharges at these stations 

 at a given instant as Qo and Q; the amplitudes of the currents as Bq 

 and B; and the initial phases of the currents as /So and j8; then; 



Qo=MoV=BoMo sin {at +^o) 

 and 



Q=Q,+1:aQ=BoMo sin (ai+i3o)+2aXM sin {at + a). (278) 



Since Q~MB sin (ai+jS), equation (278) becomes, after dividing 

 both members by M, 



5 sin {at-{-l3) = {Mo/M)Bo sin {at+^o)+{Mo/A'f)^{aU/Mo)A sin {at+a). 



The coordinate amplitudes of the current at the velocity station are 

 then: 



B sin /3= {Mo/M)Bo sm /3o+ {Mo/M)i:{aU/Mo)A sin a (279) 



B cos p = {Mo/M) Bo cos /3o + (Mq/M) S {a U/Mo) A cos a. (280) 



