APPENDIX I 



NUMERICAL SOLUTION OF STREAM FUNCTION PARAMETERS 



Introduction 



This appendix outlines the method of determining numerical values for the parameters in 

 the general form of the Stream-function solution. The numerical solution requires the use of 

 a reasonably liigh-speed, large memory computer. 

 Review of Problem Formulation 



The problem of a two-dimensional, periodic wave propagating in water of uniform depth 

 has been discussed in Section II of the main body of this report. If the water is 

 incompressible and the motion irrotational, then the following boundary value problem can 

 be established for an "arrested" wave system. 



Differential Equation (DE): 



Boundary 

 Conditions 



V^\l) = 

 Bottom Boundary Condition (BBC): 

 w = 0, z = -h 

 Kinematic Free Surface Boundary Condition (KFSBC): 



•/ 2 = n (x) 



In = _ 



9x u - C 



Dynamic Free Surface Boundary Condition (DFSBC): 



n + 1^ 1 (u - c)2 + w' 



2g 



= Q, z = n(x) 



(l-l) 



(1-2) 



(1-3) 



(14) 



Motion is periodic in x with spatial periodicity of the wavelength, L. (1-5) 



Equations (I-l through 1-5) represent the common formulation for all of the classical 

 nonlinear water wave problems in which it is assumed that the wave propagates without 

 change of form and a reference coordinate system has been chosen that travels with the 

 wave form. For a specified wave height, water depth and wave period, the goal then is to 

 determine as exactly as possible a solution to the formulation. 



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