classes as being nearly symmetric, the fractions increase to about 47 percent 

 for all cases and about 50 percent for the most common cases. This result 

 means that at least half of the distributions have sufficient asymmetry that 

 they cannot be ignored. Given the energy positioning and density errors that 

 can occur by ignoring asymmetry, as discussed above, this suggests strongly 

 that asymmetry should be included in modeling and engineering structural 

 response studies. 



Applicability of the Model of Longuet-Higgins . Cartwright. and Smith (1963) 



243. The characteristic directional distribution functions shown in 

 Figures 32 to 44 are in the form of empirical results. Ideally, these results 

 would be fitted with analytic functions so they could be easily incorporated 

 in process models and so that better interpolation could be done for more 

 concise distribution definitions given specific spread and asymmetry para- 

 meters. While this has not yet been done, a subset of the results can be 

 compared to an existing model for directional distributions. The subset 

 consists of those classes identified as symmetric, i.e., those with 0.0 < |A| 

 < 0.1 (small asymmetry). As members of the low-asymmetry asymptote of the 

 overall group of classes, their shapes depend only on one parameter, direc- 

 tional spread. These symmetric shapes can be compared to the one-parameter 

 model proposed by Longuet-Higgins , Cartwright, and Smith (1963) and defined by 

 Equations 31 and 32. 



244. Figure 45 illustrates the shapes of several members of this class 

 of function. In this model, directional spread is determined indirectly by 

 the parameter s . As s increases, the spread becomes smaller. Heights of 

 the distribution peaks necessarily become larger with increasing s to 

 satisfy the constraint of unit area under the distribution. To see if this 

 class of function can represent the present empirical observations, curves are 

 fitted to the data in each spread class by varying the parameter s . Least 

 square difference between curve and data is used as the fitting criterion. 



245. Figure 46 illustrates the results of this fitting procedure. One 

 subplot is shown for each of the 13 classes of directional spread. Data mean 

 values are shown as solid circles spaced at 2-deg intervals. Variations of 

 one standard deviation of the data about its mean values are shown as dashed 



118 



