In the formulation presented, no requirements have been placed on the permanence of 

 wave form; that is, the wave could change form as it propagates due to the relative motion 

 and interference of components propagating with various phase speeds. The treatment of 

 this general problem including the nonlinearities is complex, and was not the subject of this 

 research. Rather, in the present investigation, it is assumed that the wave propagates with 

 constant speed, C, and without change of form. It is then possible to choose a coordinate 

 system propagating with the speed of and in the same direction as the wave, and relative to 

 this coordinate system the motion does not change, and is therefore steady. The time 

 dependency in the formulation vanishes, the horizontal velocity component with respect to 

 the moving coordinate system is u-C; and the formulation may be summarized as: 



'dE: V^cf) = V^ip = (9) 



BBC: w = 0, z = -h (10) 



Boundary 

 Conditions 



i KFSBC: 1^ = ^r^' 2 = n (X) (11) 



DFSBC:n + j^ ((u-C)' + w') - |1 = Q, z = n(x) (12) 



Motion is periodic in x with spatial periodicity of the wavelength, L. (13) 



To avoid misimpressions about the assumptions and formulation presented here and those 

 employed in other investigations of nonlear waves, it is noted that the formulation 

 incorporating the assumption of propagation without change of form is common to the 

 development of all the following nonlinear water wave theories: 



Stokes 2"", and higher order wave theories 



Cnoidal 1st and 2^d order theories by e.g., Keulegan and Patterson (1940), 



and Laitone (1960) 

 Sohtary wave theory, is^ order by Boussinesq (Munk, 1949) 

 Solitary wave theory, 2^d order by McCowan (Munk, 1949) 

 Stream-function wave theory by Von Schwind and Reid (1972) 



To reiterate, analytical validity will be based on the degree to which a theory satisfies the 

 boundary-value problem formulation. Equations (9) through (13). If a theory could be 

 found that provided exact agreement to the formulations, then the analytical validity would 

 be perfect. There is no guarantee that good analytical validity ensures that a theory will 

 provide a good representation of the natural phenomenon, because implicit in the 

 formulation are the assumptions that capillary and rotation forces and other effects are 

 negligible. Experimental validity will be based on the agreement between wave theories and 

 measured data. 



