Rewriting the least -squares procedure in terms of the unknowns: AL and AX(n) 



3E 

 3AL 



9E 



9AX(n) 



= 



= , n = 1 NN 



(1-16) 

 (1-17) 



Equations (1-16) and (1-17) represent a set of NN -I- 1 linear simultaneous equations in 

 terms of the NN + 1 unknowns. After each iteration, the water surface is recalculated, by 

 iteration, from Equation (1-7) and i//^ is redetermined such that 



,L 



n dx = 



(1-18) 



which can be expressed in integral form as: 



,L/2 



♦n = L 



X(n) sinh 



2-nn 

 L 



(h + n) 



2-nn 



X dx (1-19) 



where, in the computations, a Simpson's rule approximation to Equation (1-19) is used. 



One complete iteration comprises a simultaneous solution for AL and the AK(n)'s and a 

 redetermination of \l/^. Successive iterations involve exactly the same procedure, and the 

 iterations can be terminated when successive reductions in the error E are small. Numerical 

 instabilities can occur, especially near breaking wave conditions, and one effective procedure 

 in these cases is to apply only a fraction of the AL and AX(n)'s specified by the 

 least-squares solution. 



One final comment should be directed toward the problem of estabhshing the desired 

 wave height. Although it is possible to develop more sophisticated procedures which 

 converge on the wave height, the procedure followed here was simply to conduct successive 

 runs until the wave height was within an acceptable limit (1 percent) of the desired height. 



102 



