Results of Analytical Validity Comparison 



Most of the results of the study of analytical validity carried out under this project have 

 been published elsewhere (Dean, 1968a), and therefore will be reviewed only briefly here. 



The study included 40 wave cases as shown in Figure 2. For each of these cases, the 

 overall errors, Ej and E^ were calculated for the wave theories shown in Table A. The 

 overall dynamic free surface boundary condition errors were made dimensionless by dividing 

 by the wave height, H, i.e., 



Ez' = Ez/H (20) 



The overall kinematic free surface boundary condition error is dimensionless as defined in 

 Equation (18). 



Plots of the dimensionless kinematic and dynamic free surface boundary condition errors 

 are presented in Figures 3, 4, 5, and 6 for H/Hg = 0.25 and 1.0 (H^ = breaking wave 

 height). The KFSBC error is identically zero for the Stream-function and McCowan theories. 



As stated previously, it is difficult to select a single index that would clearly be 

 representative of the overall validity of all wave theories. However, an index was chosen that 

 provided a severe test for the Stream -function theory, and yet this theory provided the best 

 general analytical validity. 



The following evaluation plan was adopted, the results of which would be somewhat 

 biased against the Stream-function theory. Most of the wave theories do not satisfy exactly 

 either the DFSBC or KFSBC; however, the Stream-function theory does satisfy exactly the 

 KFSBC. It therefore seems reasonable that if the Stream-function theory can be shown to 

 compare favorably against other theories on the basis of only the DFSBC, then it should 

 provide an even better analytical vaUdty than the comparison shows. 



In the analytical validity investigation, the eight wave theories in Table A were examined. 

 Because the fifth order was the highest of the Stokes theories available, it was arbitrarily 

 decided to include the Stream -function theory only to the fifth order. 



The evaluation was then based on comparisons presented in Figures 3,4, 5, and 6 and 

 also on the corresponding figures for H/H^ = 0.50 and 0.75, which are not presented here. 

 The results of this study are shown in Figures 7 and 8. 



Figure 7 presents the results for all theories excluding the Stream -function theory. It is 

 seen that the Stokes V theory provides the best fit for deep water, the Airy theory provides 

 the best fit in a part of the transitional and shallow-water ranges, and the first -order Cnoidal 

 wave theory generally provides the best fit in the shallow-water range. 



Figure 8 presents the same type of information. Only the fifth-order Stream -function 

 theory is included and provides the best fit over a wide range including all of the transitional 

 and deepwater wave regions and also a significant part of the shallow-water range included 

 in the comparison. The Airy wave theory provides the best fit for a small part of the 

 shallow-water, near -breaking waves and the first -order Cnoidal wave theory provides the best 

 fit for the remainder of the shallow -water region. 



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