observed directional distributions are near in shape to the analytic models. 

 However, there is no particular assurance that observations in a given 

 experiment will conform to proposed analytic models, and strong biases may 

 occur in parameterization if model and data do not conform well. 



141. Used here is a third alternative which is simple, meaningful, easy 

 to compute, and can be defined for distributions with any number of modes. If 

 the sum in Equation 14 is not taken over all directions but is truncated at a 

 particular direction 6^ , the result is the cumulative area under the 

 distribution function from 6i (= 90 deg, one of the shore-parallel limits of 

 wave direction) to the arbitrary direction 9^ . If this computation is done 

 for all values of j from 1 to M , the result is a cumulative distribution 

 function with values ranging from to 1, with the j''*^ value indicating what 

 fraction of the total energy (at frequency f„ ) is included between angles 



9i and 8. . In this sense, the directional distribution function can be 

 treated just like any statistical distribution function. One constraint on 

 this method is that wave energy must be bounded in direction. If the energy 

 is distributed over a full 360 deg (as is possible in the deep ocean) , there 

 is no meaningful starting point for the integration. In the present study, 

 the waves are assumed to arrive from a maximum range of 180 deg so that 

 integration can start from either of the two shore-parallel azimuths. 



142. The directions for which the cumulative energy is 25 percent, 

 50 percent, and 75 percent of the total have been chosen as characteristic 

 directions. Since these directions divide the energy into quarters, they can 

 be called the quartile directions . A mathematical expression for the (dis- 

 crete) cumulative distribution function is 



m=l 



'25%, n ' ■'50%, n '^"^ ''75%, n 



and the three characteristic directions, called 6-,^., „ , 8^m r, and 9- 

 respectively , satisfy the equations 



I(fn>^25%.n) = 0.25 (16a) 



I(fn.^50%,n) = 0.50 (16b) 



56 



