In evaluating the results obtained in the shallow-water region, it is noted that one 

 eighth-order Stream-function theory was calculated for breaking wave conditions and 

 h/T^ = 0.1 foot/second^ as shown in Figure 6. This figure shows that the use of higher order 

 Stream-function theories would extend the range of best vaUdity of this theory to shallower 

 conditions (Figure 8). 



Comparison with Stream-function Theory Developed by Von Schwind and Reid 

 As noted earlier, Von Schwind and Reid (1972) have developed a Stream -function 

 theory with basic similarities to the theory employed in this study. The principal difference 

 between the two theories is that Von Schwind and Reid transform their problem to and 

 carry out their solution in the complex plane. It is noted that their solution in terms of 

 wavelength and coefficients is also obtained by iteration. The DFSBC error 

 definition t^i^), used by Von Schwind and Reid was originally defined by Chappelear 

 (1961), and is somewhat different from that employed here (Equation 17) and is 



£2(6) 



It is noted by comparison of Equations (17) and (17a), that the actual distribution of 

 DFSBC errors would appear as numerically smaller based on Equation (17a) due to the 

 water depth and Bernoulli constant appearing in the denominator. 



Von Schwind and Reid presented distributed DFSBC errors for three sets of wave 

 conditions. Errors were calculated for the same wave conditions using the present theory. 

 Figures 9, 10, and II, are reproduced from Von Schwind and Reid, and the maximum 

 errors obtained by the present theory [indicated University of Florida (UF)] are shown for 

 each wave case. The maximum UF errors obtained are so small that it would not be 

 worthwhile to show them graphically. Note that aU errors (e^) shown in 

 Figures 9, 10, and 11 are based on Equation (17a). The reason that the errors obtained by 

 the present theory are smaller than those obtained by Von Schwind and Reid is not known. 

 With a numerical solution, it is possible to obtain a low error (down to some Umit) by 

 increasing the order of the theory or by increasing the number of iterations used to obtain 

 the solution. For the three cases shown in Figures 9 through 11, the UF waves were 

 seventh-order and each solution was obtained by 15 iterations; the corresponding values for 

 the Von Schwind-Reid waves are not known. 



Conclusions Resulting from the Analytical Validity Study 



The analytical vahdity evaluation is based on the degree to which the various theories 

 satisfy the governing equations in the boundary value problem formulation. It is stressed 

 again that there is no guarantee that a theory providing a good analytical vaUdity will 

 necessarily represent well the features of the natural wave phenomenon. The reason is that 

 there are assumptions (negligible viscosity and capillary effects) introduced into the 



18 



