Theory of the Tides 293 



The fluctuations above and below the disturbed mean level are given by 



r] = j i^ 7 -" cos20cos2(/tf+e) (IX. 14) 



The tide is semi-diurnal and if, as in the actual case of the earth, c < nR, 

 there will be high water at latitudes above 45°, and low water at latitudes 

 below 45°, when the moon is in the meridian of the canal and vice versa. 

 These circumstances would all be reversed if the moon is 90° from that 

 meridian. 

 (b) Canals of Limited Extent * 



In the case of a canal which does not extend over the entire earth, but 

 is of limited extent, there is no more exact agreement or exact opposition 

 between the tidal force and tidal elevation. The boundary condition 1=0 

 for the ends of the canal X = ±a is added to the equation of motion. In the 

 case of an equatorial canal, the equation of motion has the same form 

 as (IX. 11). e can be left out in its cos term if the origin of time is the passage 

 of the moon through the meridian in the center of the canal (X = 0). 



If we neglect the inertia of the water (d 2 £/dt 2 = 0), we find 



rj = ~//{cos2(/7? + ;0- S1 ^cos2///j . (IX. 15) 



This solution is designated as the elevations on the "corrected equilibrium 

 theory'\ At the centre of the canal (X = 0) we have 



rj =lH(l-~ a \cos2nt. (IX. 16) 



If a is small, the range is here very small, but there is not a node in the 

 absolute sense of the term. The time of high water coincides here with the 

 transit of the moon through the meridian. At the ends of the canal X = ±_a 

 we have according to Lamb and Swain (1915) 



rj = ±HR cos2(}it±aTe ) ; (IX.17) 



* Let us take the case of forced oscillations in a canal of small dimensions closed at both ends 

 (inland sea) generated by a uniform horizontal force X= mcos((Tt-\-e). The lake which is out- 

 stretched in a west-east direction has a length /; v = T t : T K (ratio of the period of the free oscilla- 

 tions to the period of the force) and (p the geographical latitude, then the range at the end of the 

 land-locked sea is then 



/?i tan n 



2t] = — /cos <f> v . 



8 arc 2 



For the semi-diurnal tide, the phase (referred to the centre meridian of the lake) is at the west 

 end 9/;, at the east end 3/?, if v < 1 (direct tides); with v > 1 we have the inverted (indirect) tides, 

 (see Lamb, 1932, p. 226). 



