Pn'^' = 2 D(f„,(?^'^^) de + I D(f„,^i) de + ^ DCf^,^^'^^) d9 (24) 



i=r+l 



where this must be applied to each of a total of K modes. If there is only 

 one mode, then K = 1 and P^-"-^ = 1 , since this case makes Equation 24 

 virtually identical to Equation 14. In all cases, 



I P^'^ = 1 (25) 



k=l 



for each f^ to satisfy Equation 14, i.e., that the sum of all the parts is 

 unity. 



163. To be consistent with parameterizations of the bulk and frequency 

 distributions of directional energy, there must be a set of characteristic 

 directions which indicate the peak and quartile directions in a mode, as well 

 as the mode bounding directions. A modal peak direction, given the symbol 

 ^p'^^ , is defined as the direction of the peak in D{f^,9^) [or S(f^,6^) ] 

 for mode k at frequency f^ , where the search is conducted between the 

 bounding directions O^^l and d^^l . Examples of modal peak directions are 

 illustrated in Figure 9a. 



164. Within a mode, characteristic directions can be defined at which 

 the cumulative distribution has values of 25, 50, and 75 percent of the total 

 area, just as was done for the whole distribution at each frequency and which 

 led to Equations 15 and 16. Here, the directional distribution D(fj,,^^) 

 must be normalized by P^'^^ so that the integral between the mode bounding 

 directions will equal unity. The cumulative distribution function for mode k 

 is defined by 



I'''(fn.^a) = -^ \ 2 DCfn.^^":^) d^ + f D(fn.^i) d^ 1 (26) 



•tjj L i=r+l J 



165. Figure 9b illustrates two cumulative distribution functions 

 resulting from Equation 26 for a bimodal (K = 2) directional distribution. 

 The modal quartile directions, symbolized by ^25% n > ^50% n > ^^d ^75^ „ , 

 respectively, satisfy the relations 



65 



