﻿7oS Prof. H. Lamb and Miss L. Swain on 



favoured no doubt by the prominent part which was being 

 assigned to tidal friction in various cosmical theories. 



Qualitatively there is of course nothing to be said against 

 Airy's explanation. The chief example considered by him, 

 viz. j that of an equatorial canal encircling the globe, is 

 merely a particular case of the now familiar theory of forced 

 oscillations with damping. If <$> be a normal coordinate of 

 a dynamical system we have, on the simplest assumption as 

 to the nature of the dissipative forces, an equation of the type 



^ + ^+/4 = 3> (1) 



For the free oscillations 



<£ = Ae-' T cos(CTf + e), (2) 



. . . (3) 



.... (4) 



where 





T = 2/i, 



<r- 



-v^-w 



whilst if 







<s>=c 



cos pt 



we have 



the for 



ced oscillation 









#= 



c 



= g-cos(jrt-/3) 



(5) 



to "/g= .:'-»= .„,..,: i m - • • • ( 7 ) 



provided 



Rcos j3=fjb—p 2 , R sin j3=kp. . . . (G) 



There is here a retardation of phase given bv 



_ }p = ^ 



/*— p 2 pr(ji/p 2 — l) 



In the tidal problem p = 2n, where n is the moon's angular 

 velocity relative to the rotating earth. 



The question remains, however, whether the frictional 

 forces which are operative are sufficient to account for the 

 observed differences. It was pointed out by Helmholtz * in 

 1888 that the influence of viscosity on large-scale motions of 

 the atmosphere must be absolutely insignificant, and it was 

 easy to infer that the same conclusion must hold a fortiori as 

 regards tidal oscillations of water j", where the kinematic 

 viscosity is much less. This point was afterwards fully 

 developed by Hough J, who showed in particular that with 

 even so moderate a depth as 200 metres the modulus of 

 decay (t) of free tidal motions of semidiurnal type would 

 be at least three years. It may indeed be urged that in 

 places where the tides are greatly exaggerated, as in narrow 

 channels and estuaries, there may be turbulent motions with 

 a local dissipation of energy far exceeding what takes place 



* Berl. Sltzb., May 31, 1888; Wiss. Abh. Bd. iii. p. 292. 



f ' Hydrodynamics/ 2nd ed. (1895) p. 543. 



% Proc. Loiid. Math. Soc. vol. xxviii. p. 287 (1896). 



