﻿a Tidal Problem, 739 



in the smooth regular motions postulated in the calculations 

 referred to, and it is possible that such tidal retardation of: 

 the earth's rotation as is taking place under present condi- 

 tions ma j be mainly due to this cause. Jt may fairly be 

 assumed, however, as a matter of physical intuition, apnrt 

 from calculation, that the damping of free oscillations ol the 

 ocean of semidiurnal type would hardly be sensible until 

 after the lapse of a considerable number of periods. 



If this be granted, it follows from (7) that the phase-differ- 

 ence produced by friction in an endless equatorial canal would 

 be insignificant. With such depths as occur in the ocean //, 

 is considerably less than p 2 , and tan /3 is therefore comparable 

 in absolute value with T/r, where T( = 27r/p) is the period of 

 the forced oscillation. The modulus of decay being assumed 

 to be large compared with 12 hours, ft must differ very little 

 from 180°. A phase-difference of 90°, such as is postulated 

 in some numerical illustrations of the theory of tidal friction, 

 could only arise exceptionally, by " resonance/'' in the case 

 of finite areas of water having a free period very closely in 

 accordance with the forced period. 



It seems clear that the influence of friction on ordinary 

 tidal phenomena is unimportant. It was pointed out by 

 Hough in the paper referred to that phase-differences must 

 arise in another way, from the causes indicated by Newton, 

 in limited canals or oceans*. He remarks also that an 

 example in illustration of this is furnished by the problem of 

 the finite canal which had been treated, but not fully 

 examined, by Airy himself. 



Moreover, it should not be overlooked that a mere equi- 

 librium theory, when " corrected " on the principles explained 

 by Thomson and Tait, would also give differences of phase t- 

 Consider for example the case of a canal a few degrees 

 in length lying along the equator. When the moon (or 

 antimoon) is in the zenith the differential changes of level 

 are everywhere slight, the disturbing force being nearly 

 vertical and uniform. When the moon is on the horizon, 

 the changes are again slight, since moon and antimoon now 

 nearly counteract one another as regards the horizontal force. 

 Hence at the ends of the canal there will be high or low 

 water for some intermediate position] the theory shows in 

 fact that the corresponding hour-angle is 45°. At the centre 

 the range is comparatively small, and high water coincides 

 with the moon's (or antimoon's) transit. 



* In seas whose breadth as well as length lias to be taken into account 

 the question is further complicated by the " gyrostatic " eflecl of the 

 earth's rotation. 



| Thomson and Tmt, Art. 810; k Hydrodynamics,' 3rd. ed. -p 341, 



3 B 2 



