174 



order (n) 



e" 



KFSBC error 



DFSBC error 



1 



1 X 10-1 



1.92 X 10-2 



1.03 X 10-2 



2 



1 X 10-2 



3.43 X 10-2 



1.14 X 10-3 



3 



1 X 10-3 



1.29 X 10-3 



9.33 X 10--* 



4 



1 X 10-^ 



3.42 X 10"^ 



1.40 X 10-^ 



Table A. 7: RMS errors in free surface boundary conditions for a short-crested sea 



errors in the governing equations resulting from the solution. Fenton's method is a 

 particularly powerful method for examining these errors, in that it considers an entire 

 wavelength and period at once and gives an estimation of the order of accuracy of the 

 solution. A simpler approach can also be useful and be universally applied. In the 

 case of Ohyama et al.'s method, the field equation and bottom boundary condition 

 are exactly satisfied at all orders of approximation. If the solution is accurate, the 

 errors in the free surface boundary conditions will decrease with each increase in the 

 order of approximation. While this method does not guarantee that the solution is 

 correct, most errors in the algebra or the computational code will result in a decrease 

 in the accuracy of the result at the order in which the error occurs. 



Table A. 7 contains the errors in the free surface boundary conditions at orders 

 one through four for a short crested sea. This sea is created by two waves of the same 

 height and wavelength (e = 0.1, kh = 100) intersecting at an angle of 30 degrees. 

 The values given are the root mean square of the errors computed by averaging over 

 a grid of points one wavelength in both horizontal directions, and over one period. In 

 this case the errors in the free surface boundary conditions decrease with increasing 

 order of solution. It should be noted, however, that the errors do not decrease as 

 much as e". This might be partially due to the fact that the frequency is expressed to 

 only second order. Including the fourth order frequency would probably reduce the 

 error at fourth order. This behavior warrants closer examination, particularly if the 

 method were to be expanded to higher order. 



