340 TIDAL WAVES. [CHAP. VIII 



where # refers to the proper asymptote of the curve (iv). This gives the 

 1 speed ' of any free mode in terms of that of the corresponding mode when 

 there is no rotation. 



204. As a sufficient example of forced oscillations we may 

 assume 



where the value of <r is now prescribed. 



This makes V 2 f =0, and the equation (4) then gives 



? = 4/ \ K > r ) e% a e (*") 



where A is to be determined by the boundary-condition (7), 

 whence 



fi+) 



.C (19). 



d ,. , , , 2sn ,. , 

 a faf (*>*) + /(*> a 



This becomes very great when the frequency of the disturbance 

 is nearly coincident with that of a free mode of corresponding 

 type. 



From the point of view of tidal theory the most interesting cases are 

 those of 5 = 1 with o-=n, and s = 2 with <r = 2?i, respectively. These would 

 represent the diurnal and semidiurnal tides due to a distant disturbing body 

 whose proper motion may be neglected in comparison with the rotation n. 



In the case of *=1 we have a uniform horizontal disturbing force. 

 Putting, in addition, a = n, we find without difficulty that the amplitude of 

 the tide-elevation at the edge (r = a) of the basin has to its 'equilibrium-value' 

 the ratio 



,:N 



"' 



where z=$J(3p). With the help of Lodge's tables we find that this ratio has 

 the values 



1-000, -638, -396, 



forj3= 0, 12, 48, respectively. 



When <r = 2n, we have *=0, /,(*, r) = r*, and thence, by (17), (18), (19), 



f=C ....................................... (ii), 



i.e. the tidal elevation has exactly the equilibrium -value. 



This remarkable result can be obtained in a more general manner; it 

 holds whenever the disturbing force is of the type 



f= X (r)e*<** + * + '> .............................. (iii), 



provided the depth h be a function of r only. If we revert to the equations 



