Let: 



[i] = 2^(A^ + S^)I(A^ + 6^)2 - 1] , i = 1,2,3 . (C-15) 



The expanded expression for s-mj > after collecting terms in cos kQ^- 

 and sin kQj , is : 



fAi sin Tr62 cos (4)j + i^&i) 

 Si^j = cos kfi^- 



[1] 



A2 sin Tr52 cos ((|)2 + fr62) 







[2] 











A3 



sin 



TT63 cos 



(*3 



+ 



tt6 



33 







[3] 











'Al 



sin 



TT62 sin 



C*l 



+ 



Tr6 



1) 



+ sin kS]^ , ^ ^ 



^ II] 



A2 sin 1:62 sin((|)2 + ''162) 

 __ 



A3 sin V&2, sin((()3 + irfij) 

 [3] 



and similarly for b^^- : 



b^^- = sin kfi 



- cos kS^ 





|A: 



sin 



TtSj C0S((})2 



+ 



TT-Si) 



^J 



i 





[1] 







+ 



A2 



sin 



1:62 cos {^2 



+ 



7162) 







[2] 









A3 



sin 



1163 cos((f)3 



+ 



7763) 1 









[3] 





J 





(Al 



sin 



7762 sin(4)2 



+ 



^62) 



'J ' 



1 





[1] 









A2 



sin 



1:62 sin((J)2 



+ 



7762) 









[2] 









A3 



sin 



1163 sin((t)3 



+ 



7763)] 



[3] 



For every narrow-banded wave train, | A^- | < 3 in equation (C-15), 

 unpredictable terms are introduced in the expressions for the coefficients 



