I(fn.^75%.n) = 0.75 (16c) 



143. Subscripts for the representative directions carry the frequency 

 index n since they are associated with the directional distribution at a 

 particular frequency. The function I(f„,^j) is interpolated where necessary 

 to satisfy Equations 16(a-c) since it is defined only at discrete angles. 



144. Figure 8 illustrates the relationships between the directional 

 distribution function, the cumulative distribution function, and the represen- 

 tative directions. Figure 8a shows DCf^.^^) for f„ = 0.083 Hz and for the 

 particular date and time shown in the figure. Points at discrete directions 

 have been connected by straight lines to simulate a continuous distribution. 

 Figure 8b shows the corresponding cumulative distribution function from 

 Equation 15. The horizontal dashed lines in Figure 8b show the 25 percent, 



50 percent, and 75 percent values of l(f^,6^) . Their intersections with the 

 solid curve define the corresponding characteristic angles. These are shown 

 as vertical dashed lines in both parts of Figure 8. 



145 . There are three useful parameters that can be determined from 

 these three characteristic angles. One is a directional spreading parameter 

 Ad^ which is identified as the arc which encompasses the central 50 percent 

 of the energy at frequency f^ . Mathematically, it has the definition 



A^n = ^25%,n ' ^75%,n (17) 



where it should be noted that the definition is consistent with the direction 

 axis shown in Figure 8, with angle decreasing from left to right. The spread 

 angle is shown in Figure 8 as the horizontal, double-ended arrow. It includes 

 the central 50 percent of the energy because 25 percent of the energy is to 

 the left of ^25%,n ^^'^ 25 percent is to the right of 8^^^ ^ . 



146. This parameter can represent directional spread for all distribu- 

 tions. If the directional distribution is a spike (unidirectional waves), the 

 integral is a step function and A6 -» deg . If the directional distribution 

 is white (the same energy in all directions) , then A^ -^ 90 deg , a meaningful 

 representation of the central 50 percent of the energy. If the spectrum is 

 multimodal at frequency fj^ , the central 50 percent of energy is still 

 characterized. Note that the summation of Equation 15 could be reversed, 

 starting at the other longshore azimuth to obtain the same essential result. 



57 



