294 Theory of the Tides 



in which 



„ , sin4a . cos4a— 1 



A cose = 1 ~ A — and A sin2£ = - A . 



4a 4a 



If a is small, then 



R =2a and e = -£*;+ fa (IX. 18) 



approximately apply. 



High water starts at the eastern end at the time (— a — e )/15°h, with small a, 

 therefore, at the time (9 h — a°/45°) at the western end, on the contrary, at 

 the time +(a-e )/15°, with smaller a at the time (3 h + a°/45°) h. If the canal 

 is of finite length, these times are reduced to about 9 h at the eastern end 

 and 3 h at the western end, and the tide has the simple character of a standing 

 wave, which character it loses more and more the longer the canal is. 



Taking the inertia of the water into account, it is easy to find that the 

 tide in the canal will be 



H 1 



2 p 2 -\ 



cos 2 (/zf-M) — 7~ — {sin 2 (/?/ + a) cos 2p (A + a) 

 — s\n2{nt— a)cos2/>(A — a)} 



(IX. 19) 



If p tends to the limit we obtain (IX. 15) of the equilibrium theory. 

 In all cases which are at all comparable with oceanic conditions p is con- 

 siderably greater than unity. At the ends of the canal we find 



tj = iHR 1 cos2(nt±a =F e a ) , 



where 



^ » sin 4a — sin4»a 



R 1 cos2e 1 = v^ — — . * 



and 



tf lSin2£l = P(cos4 / ,«-cos4a) 

 (p 2 — l)sin4/?a 



When a is small (IX. 18) applies again, as with the corrected equilibrium 

 theory. The value of R 1 becomes infinite (resonance) in (IX. 20), also when 

 s'm4pa = 0. This determines the critical lengths of the canal, for which there 

 is a free period of the canal equal to n/n, i.e. 12 lunar hours or half a lunar 

 day. In fact, the free oscillation period of a canal of the extension 2a is 

 4aR/c, and if this should be 



ji/n , 4pa = it . 

 If the canal is not at the equator, but in the geographical latitude <f>, the 



