172 



Fourier Component, (k) 







1 



2 



3 



4 



5 



6 



Order of KFSBC error 



4.0 



4.0 



4.0 



4.0 



4.0 



4.0 



4.0 



Order of DFSBC error 



4.0 



3.9 



4.0 



4.1 



4.0 



4.0 



4.0 



Table A. 3: Order of errors for a single wave computed to order 3 



Fourier Component, (k) 







1 



2 



3 



4 



5 



6 



Order of KFSBC error 



5.0 



5.0 



5.0 



5.0 



5.0 



5.0 



5.0 



Order of DFSBC error 



4.8 



5.0 



5.0 



5.0 



5.0 



5.0 



5.0 



Table A. 4: Order of errors for a single wave computed to order 4 



conditions is of at least one order higher than the order used to compute the results. 

 These results serve to confirm the efficacy of the Richardson extrapolation method, 

 as well as the accuracy of the wave theory up to fourth order. 



Standing Wave 



Ohyama et al.'s method can also be used to compute a standing wave by defining 

 two waves of the same wavelength and amplitude, and opposing directions. In this 

 case, the wave is not steady, and the errors in the free surface boundary conditions 

 must be considered for a full period in time and full wavelength in space. The ex- 

 trapolation to the limit method can be extended to this case by expanding the free 

 surface errors in a two dimensional Fourier series (Newland 1993). 



CO oo 



k=0 1=0 



,27r{kx/L+lt/T) 



(A.ll) 



An estimate of the order of the errors can then be computed in a similar manner as Eq. 

 A. 10. The results of this computation for a standing wave in deep water (kh = 100, 

 ei — 0.01, £2 = 0.02) are given in Tables A. 5 and A. 6. There are no values given 

 for the zeroth order in time of the kinematic boundary condition because C{e)kfi for 

 both values of e are zero to the precision of the calculation, so the values computed 

 are spurious. These results indicate that Ohyama et al.'s method is also accurate to 



