Miles 



VI. CIRCULAR HARBOR 



The eigenfunctlons determined by (2. 9) for a circular harbor 

 of radius R are given by 



^^ (r,e) = a'^^[ JoU^;,)]'' JoUdsr/R) (m = 0), 



(6.1a) 





COS m9 

 sin ni6 



(m> 1), 



(6.1b) 



and 



Jm(Jms) = (m = 0,1,2,...; s = , 1 , 2, . . . ) , (6.1c) 



where r is the polar radius measured from the center of the harbor, 

 9 is the polar angle measured from the midplane of the mouth, we 

 write 4j^^r,0) in place of i|^n(x,y), the indices m (the number of 

 azimuthal nodes) and s (the number of radial nodes) jointly replace 

 the single index n in §2, and the eigenfunctlons obtained by choosing 

 the alternatives cos m0 and sin m0 are distinct. The eigenvalues 

 are given by 



'^ms" '""Ums) • 



(6.2) 



The zero'th naode of (2. 10) corresponds to m = s = 0, for which 

 We specify M by R = 1 and - \Qh^< 6 < iO^, where 



e^=a/R«l, (6.3) 



by virtue of which we may neglect the curvature of the harbor 

 boundary over Its Intersection with the straight coastline. The 

 essential approximation Is sin 20^^ 2®M' "which Is In error by less 

 than 5% for a/R < 1. Carrying out the calculation of fx,,, (2.15), 

 and A^, (4.6), on the basis of the approximations (6.3) and (3.5), 

 we obtain 



I 



Ftms= (2 - SoJ[ 1 - (m/j;/]'' [ 1 + 0(m'e^] 



(6.4) 



for the cos m0 modes and \i-m6~ ^ ^^^ ^^® ^'^^ "^® modes [the 

 approximation (6.4) Is not uniformly valid as m — ^ oo , but It suffices 

 for all but the calculation of AJ and 



110 



