512 The Tides in Relation to Geophysical and Cosmic Problems 



3. Frictional effects of tidal currents and their relation to cosmic problems 



In a number of interesting cosmic problems, the frictional effects of the 

 tidal currents play an important role. At the time the masses of the earth and 

 of the moon were still fluid, this part was a very important one ; when the moon 

 was nearer to the earth, the tide produced by the attraction of the earth was 

 very large. Every flood protuberance exercises, however, a frictional effect 



19 I 20 I 



• Water gauge 



x Horizontal pendulum 



Fig. 211. Action of the oceanic tides in the tilting of the earth crust in the vicinity of the 



Tokyo coast (Takahasi). 



on the rotating body and both protuberances try to reduce the rotation of 

 the celestial body, acting like brake shoes on a spinning wheel. The friction 

 tries constantly to drag the flood protuberances along with the rotation of 

 the moon, so that the line connecting the two fictitious bodies (moon, anti- 

 moon) will precede the protuberances, while the two flood protuberances 

 will retard the rotation. A more accurate picture will be obtained by com- 

 puting the tides forced by a tide generating force in an equatorial canal when 

 there exists a frictional effect. The case without friction has been discussed 

 in Chapter IX, 3, p. 290. The equation of motion will be modified by intro- 

 ducing at the right a frictional term —P(dg/dt) in which the frictional effect 

 has been taken proportional to the velocity of the current. We find (see 

 Lamb, 1932, § 319, p. 565) that to the horizontal tidal disturbing force 



X = — wsin2(/?? + A + £) 

 correspond a surface elevation. 

 H 1 



n 



2 I # 2 « 2 tf 4 l 1/2 



r p)+ Ac* \ 



cos2(«/ + A + e— x) , 



(XV. 7) 



in which p 



with 



nRjc and % is given by 



1 (5na 2 



tan 2 % 



2 c 2 1-p 2 



(XV. 8) 



o <x < 



