1/2 



LO- 



2kh > 

 sinh (2kh) 



(3D 



tanh (kh) 



The dispersion relation, Snell's law, and this simple estimator of the wave 

 height allow an initial guess to be made for the variables of interest 

 throughout the grid system. 



26. The solution scheme implements the following marching procedure 

 once initial guesses for the variables of interest have been made. Starting 

 at the offshore row designated by i=M-3 , Equations 21 and 26 are used to 

 compute wave angles and then heights along the entire row (from j=2 to 

 j=N-1). Wave height is used interchangeably with amplitude function since 

 one is directly proportional to the other. 



27. Wave angle and height solutions along a given row are solved 

 iteratively because of the implicit differencing formulation used. Calcula- 

 tions of the wave angle (actually the sine of the wave angle) and the wave am- 

 plitude function are reiterated until the average change (along a row) in each 

 variable from one iteration to the next is less than some tolerance. These 

 convergence criteria, 0.0005 for wave sines and 0.001 ft* (or a metric equiva- 

 lent) for wave heights, are suggested values for prototype applications. They 

 can be easily changed by modifying the source code using the method outlined 

 in Part IV. 



28. This solution considers only refraction since the wave number k 



is used as an estimate of the magnitude of the phase function gradient. Equa- 

 tion 14 is then used to compute the true magnitude of the wave phase gradient. 

 This new wave number accounts for the effects of diffraction. Backwards dif- 

 ferences are used to approximate the x-derivatives because they require only 

 information which has already been computed. Next, Equations 21 and 26 are 

 again solved in order to compute the wave angles and heights using these new 

 wave numbers. This procedure is repeated along the row under consideration 

 until the change in new wave number, from one iteration to the next, is less 

 than 0.5 percent of the newly computed value. This condition must be met at 

 each cell along the row. As a row of new wave numbers is computed, the values 

 are filtered in the y-direction using the method of Sheng, Segur, and Lewellen 



* To convert feet to metres use a conversion factor of 0.3048. 



IT 



