Stream-function Solution 



The general form of the Stream -function solution is: 



NN 



i|^(x,z) = I z -H ^ x(n) sinh [i^l (h + z)] cos [^ x] a-6) 



The water displacement, 7?, is determined by setting z = tj in Equation (1-6) 



NN , ^ r ^ 



^ = I '^n " r I ^^^^ ^^^^ [^ (^ ^ ^)J ^°^ (^ ^J (l-'^) 



where i//„ is the (constant) value of the Stream-function on the free surface. The velocity 

 components are defined by: 



u - c = - II a^) 



w 



In continiiing the quest to determine a solution that satisfies Equations (I-l) to (1-5) as 

 faithfully as possible, it is noted that for arbitrary values of: \Ij^, L, and the X(n)'s, the 

 Stream-function solution exactly satisfies aU of the requirements of the formulation except 

 the DFSBC, Equation (1-4). AU of the effort can therefore be directed to determining these 

 "free" variables such that they represent the specified wave height and also "best" satisfy 

 Equation (1-4). The approach employed is numerical iteration, in which a trial solution is 

 regarded as available and at each step of the iteration; the "free" variables are modified to 

 improve the solution. 



As a preliminary step, an error is defined in the one remaining unsatisfied boundary 

 condition, 



E = i I (Q. - Q)' (MO) 



J j=i : 



where the Q's represent equally spaced (in 6) values of the quantity in Equation (14), 

 and Q represents the average of the Q's. If, for example, J = 41, and the free variables 



100 



