J n (kr 2 ) Y n (kri) - J n (k ri ) Y n (kr 2 ) = . (90) 



The terms J and Y are Bessel functions of the first and second 

 kind, respectively. The order is given by the integer subscript and 

 the superscript refers to differentiation with respect to the argu- 

 ment. The term A in equation (88) is an arbitrary constant repre- 

 senting the initial amplitude. The first azimuthal mode is for 

 n = 2 and the lowest value of k that satisfies equation (90) is 

 approximately 1.340/rj meters -1 . The period of oscillation for this 

 mode (m = , n = 2) is 25.85 hours. 



The numerical solution of the free wave in the quarter annulus 

 is sought by performing the integration in two different computing 

 grids. In one case the grid is rectilinear (Figure 51), and in the 

 other a polar system is used. In the rectilinear system the outer 

 and inner radii of the annulus (the light line in Figure 51) is simu- 

 lated in a stairstep fashion. Considering a limitation on computer 

 time and storage, an acceptable rendition of the curved boundaries 

 is present in the Cartesian grid. Proper representation of the 

 quarter-annulus requires a transport point at corners on the outer 

 and inner boundari-es. Consequently, the rectilinear boundary is not 

 symmetric about tt/4. The locations of nine hydrograph positions are 

 indicated by small boxes. Although the computing grid is 43 by 43, 

 only 1,052 points are used to represent the annulus. In this grid 

 system, AS* and AT* are just Ax and Ay , respectively, with 

 Ax = Ay = 19.65 kilometers. The maximum allowable time step as 

 determined from equation (62) is approximately 700 seconds where F = 

 p = v = 1. Finally, the analog form of the long wave equations is 

 obtained by setting those terms in equations (55) through (57) to 

 zero which are neglected, and setting F , u , and v to one. The 

 transports, Qn* , Q T * , are now aliases for Q and Q , 

 respectively. 



The numerical algorithm for the rectilinear grid system must be 

 capable of determining if a computational point is interior, 

 exterior, or on the boundary of the annulus. For exterior points, no 

 calculations are performed. Grid points on the stairstep boundary 

 require special attention subject to the condition of a wall. 

 Furthermore, the numerical program must identify and apply the appro- 

 priate wall condition depending on whether the point in question is 

 located at a boundary corner (an H computation) or on a segment of 

 the boundary. Clearly, extensive programing and computer time is 

 required to accomplish this task. 



The other computing mesh which is used for obtaining the numerical 

 solution of the free wave in the annulus is the polar (or stretched 

 shelf coordinate) grid system (Figure 52) . The transformation 

 to the computing grid as shown in Figure 53 is accomplished 



92 



