The Stream-function Solution 



For the formulation expressed in Equations (9) through (13), a Stream-function solution 

 may be expressed as: 



4)(x,z) =k z +~ I x(n) sinh [^p (^ + z)J cos fe x] (14) 



Evaluating this expression on the free surface, i.e., setting z = 7?, we find 



n = J ip^ - I T X(n) sinh [2p (h + n)) COS [^ x) 



(15) 



where NN represents the order of the representation, i.e., the number of terms 

 contributing to the series expression, i//_ represents the (constant) value of the 

 Stream-function on the free surface, L is the (undetermined) wavelength, and 

 the X(n) represent, at this stage, undetermined coefficients. 



For particular wave conditions, it is regarded that the wave height, period, and water 

 depth are specified. Equation (14) exactly satisfies the governing differential equation and 

 the bottom and free surface kinematic boundary conditions for arbitrary values 

 of L, i//^ and the X(n) coefficients. The Stream-function expression is also periodic 

 in X with wavelength, L. The only remaining boundary condition is the dynamic 

 free-surface boundary condition; the parameters L and the X(n)'s are to be chosen such 

 that this boundary condition is best satisfied for a specified wave height. 



The procedure for determining the unknown parameters, which can be considered as a 

 nonlinear numerical perturbation procedure, is presented in Appendix I. 



m. EVALUATION OF VALIDITIES OF WAVE THEORIES 

 Introduction 



As discussed earlier, there are two types of validity that were examined. "Analytical 

 validity" is based on the degree to which a theory satisfies the governing equations (of the 

 boundary value problem formulation). Good analytical validity, however, does not 

 necessarily imply good representation of the natural phenomenon. Experimental validity is 

 based on the agreement between a theory and measurements. To date, some reasonably 

 good laboratory data are available, and at least two field measurements of water particle 

 velocities are reportedly underway (as of 1972) in the petroleum industry, and hopefully, 

 will be available within the next few years. 



Discussion of Differences Between Stream-function and Other Wave Theories 



Later in tliis section, it will be shown that the Stream-function theory provides a better 

 fit than other theories to the boundary conditions and also provides a better fit to 



