and 



1 ) 1/2 



r ouu 1 > (37) 



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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. 



37. 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 31 and 32 are used to 

 compute wave angles and then heights along the entire row (from j=2 to j=N-l). 

 Wave height is used interchangeably with amplitude function since one is 

 directly proportional to the other. 



38. Wave angles and heights along a given row are solved for itera- 

 tlvely because of the implicit differencing formulation used. Calculations of 

 the wave angle (actually the sine of the wave angle) and the wave amplitude 

 function are repeated 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 sines of the wave angles and 0.001 ft (or a metric equiv- 

 alent) for wave heights, are suggested values for prototype applications. 



39. 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 27 is then used to compute the true magnitude of the wave phase gradient. 

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

 differences are used to approximate the x-derivatives because they only re- 

 quire information which has already been computed. Next, Equations 31 and 32 

 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 considera- 

 tion 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 (1978). This filter removes cell-to-cell oscillations introduced as 

 a result of the differencing scheme used to compute the new wave numbers. 

 Row-by-row marching proceeds until solutions are computed along row 1=2. 



27 



