For example, the combination 1, 5^ 1 , 10 in table 3 accounts for 

 61.96 '^^ an increase of 1.27 ^ over combination 1, 5^ 7» In table G, the 

 combination 8, 9^ 13 accounts for 0-77 %, and the combination 8, 9, 10, 

 13 accounts for 1.88 % an increase of 1.10 .^. Thus the position of XIO 

 depends very largely on the strength of the combination in which it occurs, 

 and XIO seems to play a minor role in this set of data in that it first 

 shows up as weak or strong in combinations of h Xs at a time. It will be 

 recalled from table 2 that XIO accounts for I.I9 ^ of the SS of Y, a con- 

 tribution that seems to remain much the same throughout the analysis. 



Simimary 



Based on an analysis involving eleven independent variables, it is 

 possible to say that the combination of six factors (table 5) that is most 

 influential in the determination of longshore- current velocity in the study 

 area at Virginia Beach is the combination made up of wave period, wave 

 height, lower foreshore slope, wind velocity onshore, wind velocity offshore, 

 and angle of wave approach, in that order. (When the derived variables of 

 wave length and wave sjteepness are added to the analysis, the six most- 

 important factors are Sf , T, Lq, Hq/Lq, Uo^^ aJ^d Uq-^ ) . The wave factors 

 and the beach slope are variables agreed upon by workers in the field of 

 longshore- current generation to be of fundamental importance. The signi- 

 ficance of winds on and offshore is believed to lie in their ability to 

 alter the wave form prior to breaking. 



The least-squares relations developed by this analysis appear to 

 adequately represent the combinations of variables of most significance in 

 natiore. It is to be recalled that the framework of analysis is linear, but 

 it -is not uncommon, when a large number of variables is involved, that a ■ 

 linear approximation yields reasonable resiilts, even thougji some relations 

 may be known to be non-linear. Moreover, non-dimensional ratios among 

 some of the original, variables, such as H /Lq in this example, may rise to 

 greater relative importance than the origxnal variables themselves, as 

 shown in table 2. This suggests that further analysis by use of non-dimen- 

 sional variables alone (perhaps as derived from use of Buckingham's Pi 

 theorem) may be useful and informative. We hope to extend the analysis 

 in this direction. 



As a summary of the foregoing linear analysis, it seems fair to 

 say that the least squares techniques used here have helped "sort out" 

 the relative interplay of a gjjoup of variables as they affect longshore- 

 ciirrent velocity on a particiilar beach during a given time-span . Statis- 

 tically oiir model is "fixed", and extensions of our findings to general- 

 izations about other beaches is valid mainly in that the underlying variables 

 are perhaps the same, even though their relative rankings may vary from 

 beach to beach, or on the same beach from time to time. That non-linearity 

 is also a problem is discussed next. 



THE PROBLEM OF NON-LIEEAEITY 



The occurrence of non-linear relations among beach-ocean-atmosphere 

 variables was touched upon above, and it is discussed here in connection 



27 



