336 TIDAL WAVES. [CHAP. VIII 



In the case of uniform depth, we find 



V^f ........................ (4), 



where V i 2 = ^ 2 + ~ 4~ + ~ 2 



dr* r dr r 2 



and K = (<7*-4,n*)lgh ....................... (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 = Q, or 



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

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

 in conjunction with Fourier's theorem, suggests that 6 occurs 

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

 tion, the differential equation (4) becomes 



= (8), 



TJ ' 



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) ........................... (10), 



where 



~ 



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

 numerical factor from J S (K^T) or 7, (#'*?), where K is written 

 for /c, and I 8 (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. Rep. 

 1889. 



