laboratory measurements of water particle velocities. It is therefore worthwhile to compare 

 some of the inherent features of the Stream-function and other theories. Although it is 

 difficult to discuss all other theories in general statements, an attempt will be made to 

 present the more significant representative differences. 



Consider, as an example, the Stokes higher order wave theories. The general form of the 

 solution exactly satisfies the differential equation, the bottom boundary condition, and, is 

 properly periodic in the x-direction. The solution does not provide exact fits to either the 

 kinematic or dynamic free surface boundary conditions. Suppose that the (n-1)*" order 

 solution is known and that the nth order theory is to be developed. The n^^^ coefficients are 

 determined such that they minimize the errors in the two free surface boundary conditions 

 at the (n-l)^'i order. A significant problem is that the configuration of the n^" order water 

 surface is not known, a priori; it is therefore necessary to best satisfy the boundary 

 conditions on an approximate expansion of the n^'' order water surface. The apparent effect 

 of minimizing the errors present on the approximate n^" order water surface is that the 

 resulting theory of a given order, if convergent, may not provide the best fit possible for the 

 number of terms (order) included. 



As a comparison with the preceding discussion of the Stokes' theory, consider the 

 corresponding features of a Stream-function theory solution. The general form of the 

 solution exactly satisfies all of the boundary value problem requirements except the 

 DFSBC. 



At this stage, one inherent advantage of the Stream -function theory is evident— all of the 

 "free" parameters can be chosen to provide a best fit to the DFSBC. A second and 

 important inherent advantage is that for a given n^^ order wave theory, all of the 

 coefficients are chosen such that they best satisfy the boundary condition on the n^" order 

 water surface. The distinction is that because a numerical iteration approach is used, the nth 

 order wave form is known (through iteration) at that order of solution. Other advantages of 

 the Stream-function theory are that a solution can readily be obtained to any reasonable 

 order, and that a measure of the fit to the one remaining boundary condition is more or less 

 automatically obtained in the course of the solution. Also, the form of the terms in the 

 solution is inherently better for representing nonlinear waves, due to the 17 term appearing in 

 the argument of the hyperbolic sine term [cf. Equation (15)]. 



The disadvantage of the Stream-function theory is that, unless tabulated parameters are 

 available, it does require the use of a digital computer with a reasonably large memory. The 

 complexity of other nonlinear theories, however, generally also requires the use of a 

 high-speed computer. 



It is noted that a similar but different Stream-function theory has been developed and 

 reported by Von Schwind and Reid (1972) subsequent to the analytical validity study 

 reported here, and employs a definition of the DFSBC error which differs from that in the 

 present study. The paper by Von Schwind and Reid presents boundary condition errors for 

 three wave cases. A comparison between their errors and those resulting from the 

 Stream-function theory will be presented. 



