200-202] STRAIGHT CANAL. 335 



We thence derive 



c 2 = gh t m 2n/c ..................... (4). 



The former of these results shews that the wave-velocity is 

 unaffected by the rotation. 



When expressed in real form, the value of f is 



%=ae-y! c cos{k(ct-x)+e} ............... (5). 



The exponential factor indicates that the wave-height increases as 

 we pass from one side of the canal to the other, being least on the 

 side which is forward, in respect of the rotation. If we take 

 account of the directions of motion of a water-particle, at a crest 

 and at a trough, respectively, this result is easily seen to be in 

 accordance with the tendency pointed out in Art. 194*. 



The problem of determining the free oscillations in a rotating 

 canal of finite length, or in a rotating rectangular sheet of water, 

 has not yet been solved. 



202. We take next the case of a circular sheet of water 

 rotating about its centre f. 



If we introduce polar coordinates r, 6, and employ the symbols 

 R, to denote displacements along and perpendicular to the 

 radius vector, then since R = iaR, = iV, the equations (9) of 

 Art. 200 are equivalent to 



whilst the equation of continuity (10) becomes 



d(hRr) d(hQ) . 



rdr rd6 



TT r&amp;gt; 



Hence R 



9 







and substituting in (2) we get the differential equation in f. 



* For applications to tidal phenomena see Sir W. Thomson, Nature, t. xix. 

 pp. 154, 571 (1879). 



t The investigation which follows is a development of some indications given 

 by Lord Kelvin in the paper referred to. 



