Cartwright, and Smith (1963), at least for classes with low- to- intermediate 

 spread. This suggests that such a model might serve as a basis for an 

 extended model which includes all of the shapes found. 



261. The rather exciting consequences of these results are as follows: 



a. There is not an infinite variety of directional distribution 

 shapes. The composite curves indicate that there is a 

 distinct order in the way wave energy behaves in shallow 

 water. 



b. At least half of all distributions can be represented by a 

 simple, two-parameter mathematical function. It has yet to be 

 defined analytically; but it exists. At a minimum, the 

 empirical shapes published here can be used for modeling even 

 though they are somewhat awkward, being in discrete form. 



c. Quite probably, all of the distributions can be represented by 

 this two-parameter function. By scaling and positioning one 

 of these for each mode in a multimodal distribution, a 

 complete distribution of any degree of modality can be 

 constructed. Consequently, the shapes found in this study may 

 be used as fundamental building blocks with which to construct 

 a complete frequency- direction spectrum. This makes a 

 complete spectrum much easier to characterize than having to 

 consider energy separately at all locations in the frequency 

 and direction domains. 



d. It is evident that asymmetry in the distributions is as 

 important a parameter as directional spread, especially at the 

 larger spreads. While failure to account for it may lead to 

 serious errors in spectral interpretation and model results , 

 the fact that it has been found and quantified in the present 

 results means that this type of structure can be included in 

 future studies. 



262. Together, these results and ideas suggest that there is potential 

 to characterize and describe in detail entire classes of frequency-direction 

 spectra with just a few parameters. This would be an extremely useful result 

 for engineering design because it would allow detailed definition of all, 

 including the most destructive, wave environments to be characterized in a 

 simple way. What is striking about the present results is the strong sugges- 

 tion of order in what could potentially be a highly irregular system. 



263. Compared to what is desired for knowledge of wave directionality 

 in shallow water, the present results are clearly of the most rudimentary 

 variety. A great deal of work remains to be done. In the first place, the 

 present results need to be incorporated into the realm of characterizing the 

 detailed structure of complete frequency-direction spectra. This means 



126 



