IV. SIMULATION OF THE FREE WAVE IN AN ANNULUS 

 1. Problem Statement . 



If the wind stress, bottom friction, atmospheric pressure, and 

 the rotational effect of the earth are neglected, then the equations 

 in Section III admit a simple analytic solution for free gravity 

 waves of long wavelength. A study using curved boundaries will 

 demonstrate the superiority of modeling the long gravity wave in 

 orthogonal curvilinear coordinates over those models which employ 

 rectilinear coordinates. Basically, the study involves comparing 

 the numerical solution of a free-standing long wave in a 90° section 

 of an annulus with an inner radius r^ of 393 kilometers and an 

 outer radius r£ equal to 2r^ (Figure 50) . The annulus is bounded 

 on all sides by a vertical wall. The depth of the basin relative 

 to the mean water level is assumed everywhere to be 40 meters. 



The analytical solution for the free-wave oscillations in a 

 section of an annulus may be obtained with some modifications from 

 Lamb (art. 191, 1932). The boundary conditions require that: 



8H 



, at r = rj and r = t^ , 



(86) 



and 



3H 



= , at 6 = 0° and G 



tt/2 



(87) 



The analytical solution for the given initial conditions is 



H(r,0,t) = H A 



m,n 



J (k r u 



t ri ^ n IIl,n J v n -i 



J (k r) - — Y (k r) 



n m > n Y*(k r x ) n m ' n 

 n l m,n lJ 



cos n8 cos a t , 

 m,n 



(88) 



m = 0,1 

 n = 0,2 



where 



(89) 



and for given n , k is the m tn root of 



s ' m,n 



90 



