171 



Fourier Component, (k) 







1 



2 



3 



4 



5 



6 



Order of KFSBC error 



2.0 



2.0 



2.0 



2.0 



2.0 



2.0 



2.0 



Order of DFSBC error 



3.0 



2.2 



2.0 



2.0 



2.0 



2.0 



2.0 



Table A.l: Order of errors for a single wave computed to order 1 



Fourier Component, (k) 







1 



2 



3 



4 



5 



6 



Order of KFSBC error 



3.0 



3.0 



3.0 



3.0 



3.0 



3.0 



3.0 



Order of DFSBC error 



2.9 



3.0 



3.0 



3.0 



3.0 



3.0 



3.0 



Table A. 2: Order of errors for a single wave computed to order 2 



The magnitude of this error, and thus each of the coefficients can be expanded in a 

 first order Taylor series. 



Ckiik) = Mk + o [iD 



(A.8) 



These coeflacients are a function of the small parameter, e and thus are a function of 

 e" at nth order, ^k = e"'- 



c,(6) = /3,e"^- + o (6^"'=; 



(A.9) 



When the discreet transforms of the errors are computed for two different values of e, 

 an approximation for the rik can be computed, giving an approximation to the order 

 of the errors of each harmonic in each boundary condition. 



log(C,(62)/C,(ei)) 



nk = 



+ 0(ei,€2) 



(A.IO) 



log(e2/ei) 



Table A.l presents the results for a single wave, computed to order 1 in deep water 

 (kh = 100, ei = 0.01, 62 = 0.02). These results show that the error is of at least 

 second order in both of the boundary conditions, indicating that the computation is 

 accurate to first order. 



Tables A. 2 through A. 4 present the results for the same wave, computed to sec- 

 ond through fourth order. These results indicate that the error in both boundary 



