(11.14) A = ^c2[(k+ Y - (k-^)^] 



_ TT ^2r, 4 , 4k3 6k2 , 4k ^i , 4 x 4kl 6k^ , 4k J^, 



= u C^[2k^ +|] 



= 20.315- 10-4(2k^ +|) 



The values of A are tabulated in Table 11.4 for future reference, A 

 value of k equal to 27 corresponds to the largest circle that can be drawn in 

 the plane of the directional spectrum. The values of A increase slightly 

 more rapidly than the cube of the values of k. Also tabulated in Table 11,4 

 are the values of A divided by 36 for future reference. 



A Cartesian coordinate grid was constructed by drawing heavy lines at 

 the values of r and s corresponding to 0.5, 1,5, 2,5, ..,,, 19.5. This 

 divided the plane of the spectrum into 741 squares assigned unit area, 116 

 half squares, and 4 quarter squares for a total of 800 full squares. 



The radii given by setting k equal to 1, 2, 3, . . , , and 28 in equation 

 (11.3) were then comiputed and semicircles with these radii were superim- 

 posed on the grido 



The semi-circles divided the squares into pieces. The number and size 

 of the pieces depended on the geometry of the system. 



The areas of the pieces were then computed from geometrical consider- 

 ations which depended essentially on differences between areas of sectors of 

 circles and triangles. The squares along the r axis, the squares at 45° to 



the r axis and those in between out to the largest radius were the ones that 



were analyzed because all others could be obtained by reflection in either 



195 



