336 TIDAL WAVES. [CHAP. VIII 



In the case of uniform depth, we find 



d? 1 d 1 d? 

 where V f = ^ 2 + - ^ + - ^ .................. (5), 



and K = (&amp;lt;7 2 - 4m*)/gh ....................... (6). 



This might have been written down at once from Art. 200 (12). 



The condition to be satisfied at the boundary (r = a, say) 

 is R = 0, or 



203. In the case of the free oscillations we have f= 0. The way 

 in which the imaginary i enters into the above equations, taken 

 in conjunction with Fourier s theorem, suggests that 6 occurs 

 in the form of a factor e is& , where s is integral. On this supposi 

 tion, the differential equation (4) becomes 



dr 2 r dr 

 and the boundary-condition (7) gives 



for r = a. 



The equation (8) is of Bessel s form, except that K is not, in 

 the present problem, necessarily positive. The solution which is 

 finite for r = may be written 



r = 4/ 8 (,r) ........................... (10), 



where 



1 &quot; 



According as K is positive or negative, this differs only by a 

 numerical factor from J s (fc^r) or I s (/c *r), where K is written 

 for K, and I s (z) denotes the function obtained by making all 

 the signs + on the right-hand side of Art. 187 (4)*. 



* The functions I,(z) have been tabulated by Prof. A. Lodge, Brit. Ass. Eep. 

 1889. 



