169 



Figure 54. 



Let A B C F figure 54, be a circle 

 of radius X/27r, and hence with a cir- 

 cumference of length equal to X; and 

 let A B C be an arc of length L. 

 The subtended central angle AOC is 

 then i/(X/27r)=7 radians, and the 

 length of the chord A(7is 2(X/27r). sin Kt- 

 The ratio Vmlvi is therefore the ratio 

 of the arc ABC to the chord AC; 

 and 1/cos Kt is the ratio of OC to OD. 

 It is apparent therefore, that if the 

 length of a canal is not too large a 

 part of the wave length of the tidal 

 components, frictionless tides and cur- 

 rents at its middle do not differ greatly from the values which they 

 would have if the effect of tidal storage were neglected. 



SPECIAL CASES 



332. A canal connecting a tidal with a tideless sea. — If the sea at one 

 end of the canal is tideless, all of the components there have a zero 

 amplitude and equations (228) and (230) become: 



y=Ao cos (at+ao) sin (1— j"/Z)7/sin y (242) 



'V^(9/c)Ao sin (at-\-ao) cos (1— a:-/iy)7/sin 7 ■ (243) 



It is apparent that, when the equations reduce to this form, y be- 

 comes a maximum and v becomes zero, for all values of x, when at-\-ao = 

 0, or when t^ — aja; and that ?/— and y is a maximum, when t= 

 — Qio/a + Tr/a. Both high water and low water occur therefore at the 

 same respective instants throughout the canal, and the current turns 

 at the same instants. 



If, for example, the canal described in paragraph 328 entered a 

 tideless sea at its further end, the tides and currents at the entrances 

 and at the middle of the canal, and the instantaneous profiles at suc- 

 cessive lunar hours, would take the forms shown in figure 55, page 170, 

 were the flow frictionless. 



333. When high water occurs at the same time at both entrances, or 

 when the tides at these entrances are exactly opposite. — If the phases of 

 the tides at both entrances are the same, ai = ao, and equations (228) 

 and (230) reduce to: 



^ = cos {at + ao)[Ao sin (1— //L)7 + ^i sin (.r/Z)7]/sin 7 (244) 



v=ig/c) sin (a^+«o)[A cos (l—x/L)y—Ai cos (.r/i)7]/sin 7 (245) 



In this case, also, high water and low water each occur at the same 

 instants throughout the canal, and the currents turn throughout the 

 canal at these instants. It is readily shown that the same conditions 

 residt if the phases of the tides at the ends of the canal differ by 180°. 



