Streetj Chan and Fromm 



third- order theory Is found. 



To obtain a meaningful comparison with the profiles of the 

 Stokes' waves, a Fourier analysis was performed on the profile 

 computed by the SUMMAC method. The SUMMAC wave profile in 

 Fig. 13 can be expanded in a Fourier series of the Stokes form 

 [ Wiegel, 1964] . The coefficients can be evaluated by the standard 

 procedures in calculus. The first ten coefficients have been conti- 

 puted and connpared with those for the Stokes' theories. From the 

 trend of each coefficient, it appeared that as the order of approxi- 

 mation increases the Stokes' wave converges to our numerical 

 solution. Also, In comparison of wave speeds we find good agree- 

 ment with Stokes' third- order theory. The difference is within 

 0.4 per cent. 



In Fig, 14 the distribution of u under the wave crest and the 

 wave trough is compared with Stokes' theory. The SUMMAC method 

 predicts a much lower u velocity under the crest than Stokes' solu- 

 tions. This discrepancy Is probably caused by the fact that the 

 numerical simulation was made In a channel of finite length which Is 

 a closed system and the waves have not quite reached the steady 

 state, while the Stokes' waves hold for an Infinitely long channel. 

 Nevertheless, the slope of the u- distribution (I.e., 8u/8y) Is very 

 close to that of the third-order theory. 



o_ 





\ 



(0) 



UNDER WAVE 

 TROUGH 



< I 



JJ_L 



(b) 



UNDER WAVE 

 CREST 



STOKES' SECOND 

 ORDER THEORY 



STOKES' THIRD 



ORDER THEORY 



0.1 0.2 



Fig. 14. Distribution of u under wave crest and trough 



182 



