governing equations which may adversely affect the degree to which the formulation 

 represents real wave motion. The purpose of the analytical validity study, rather, was to 

 attempt to resolve the question of whether the theories developed for the same formulation 

 and for various regions of relative depth do indeed provide the best fit in these regions. Also 

 this study, combined with some additional studies reported later in this report, does aid in 

 determining whether the most critical need in wave theory research is in the improvement of 

 the formulation or in the development of improved solutions to the existing formulation. 

 The results of the analytical validity study have shown that: 



1. The general status of wave theories for h/t^ > 0.2 foot/second^ , for instance, is much 

 more satisfactory than for the smaller values of h/T^ . In particular, for the larger relative 

 depths, there is reasonable consistency between the fits to the dynamic free surface 

 boundary condition and the maximum drag force as calculated by the various theories 

 including a seventh-order Stream-function theory. In shallow water, it is not clear that the 

 boundary condition fit is an appropriate measure of wave theory validity, unless the 

 associated errors are very small. In particular, the Airy wave theory provides a relatively 

 good fit to the boundary conditions in shallow water; however this theory does not 

 represent many of the observed features of shallow-water waves including the strong 

 skewness of the wave profile about the mean water level. 



2. The Stokes higher order wave theories converge to accurate representations of wave 

 motion in deep water; however, in transitional and shallow water, the boundary condition 

 fits are relatively poor. Furthermore, no fifth-order Stokes theory solution could be found 

 for shallow-water waves or the smaller values of the transitional zone. The limiting value 

 of h/T^ for which a solution exists, depends on H/T^ and was in the range of 

 O.K h/T^ < 0.5 foot/second^ for the conditions examined. 



3. Finally, it is observed that the second-order Cnoidal theory provided a worse fit to the 

 boimdary conditions than the first-order Cnoidal theory for all wave conditions examined. 

 There are other versions of Cnoidal theories; the boundary condition fits of these theories 

 have not been evaluated in this study. 



4. The Stream-function theory described in this report provides good analytical validity 

 over a wide range of wave conditions. 



The reader is referred to Dean, (1968a) for reinforcement of statements presented. 



Experimental Validity 



As previously described, experimental validity is based on the comparison of theoretical 

 predictions and measured wave phenomena. If it could be generally shown that the theory 

 providing the best analytical validity also provides the best experimental validity, then it 

 could be concluded that the formulation is valid and that the errors in the boundary 

 conditions are also good indicators of experimental validity. If the differences between the 

 theory and experiments were of the same order as the estimated experimental error, and if 

 this could be shown to be the situation generally, then the most productive direction in 



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