V^alytical Validity 



The analytical validity of a particular wave theory has been previously defined as the 

 degree to which the theory satisfies the defining equations, i.e., Equations (9) through (13). 

 Again, for emphasis, it is noted that a tlieory providing an exact fit to the boundary 

 conditions would have a perfect analytical validity. However, due to assumptions of ideal 

 flow, etc., in the formulation of the problem, a perfect analytical validity does not ensure 

 that the theory would provide a good representation of laboratory or field phenomenon. 



The reason for viewing the problem in two steps, i.e., analytical and experimental 

 validity, is that the results of the analytical validity test would at least tend to indicate the 

 relative appUcability of the available wave theories for particular wave conditions. Also, the 

 results would provide guidance about whether the most fruitful approach would be directed 

 toward a more representative formulation of water-wave theories or toward the 

 improvement of the solutions of existing formulations. 



Definition of Boundary Condition Errors 



Most wave theories exactly satisfy the governing differential equation and bottom 

 boundary condition, although some of the solutions only approximately satisfy the 

 differential equation. Table A lists a number of the theories available for design use and also 

 indicates the conditions of the formulation which are satisfied exactly by each of the 

 theories. Inspection of Table A shows that the two nonlinear (free surface) boundary 

 conditions provide the best basis for assessing the analytical validity, because no theory 

 exactly satisfies both of these conditions. 



Errors based on the dynamic and kinematic free surface boundary conditions, are defined 

 as functions of phase angle (6) as follows: 



£2(6) = n + — [(u - C)^ + w^] - 9i _ Q (17) 



where Q represents the mean value of the quantity Q (Bernoulli constant) defined in 

 Equation (12). Overall errors are defined as the root mean squares of the distributed errors, 



El = Vj ^ ^^ ' =V^ (i«) 



j=l J 



J 





2 



£2 



(19) 



where j represents sampling at various (evenly spaced) phase angles. 



