— — — — — iwt , — — i(i)t 



e for the values of v and 6 at x = , and also v = v^^^e and 60 = Q ^gP 



in terms of complex amplitudes v and 6 . Then, after canceling out 



oa oa 



e , it IS found that: 



-•■-'-- - - d + q d, 

 12 2 1+ 



y ^CJ - iwc) v__^ = - q^d^ + q^d 





(Note that 3 sin q /8x = - ^1-, sin q x, = when x = 0, etc.) These two 



equations and Equations [39a,b] are easily solved for the d's with the 



following results : 



q-, (q-, + q„) d = q A - u (w - iwc) 6 

 112 12 oa 



(q^ + qg) ^2 = - q2 B - y (w^ - itoc) 7^^ 



— p -? -2 / 2 '^ \ ~~ 



q^ (q^ + q^) d^ = q^ A + y (w - iwc) 6 



-o -2 "2 2 '^ 



(q^ + ^2^ ^1+ " ~ ^1 ^ "^ ^ ^" " ^'^'^^ ^oa 



The d's can now be eliminated entirely by multiplying Equations [39c,d] 



t>y (q + q ) and then substituting in these equations the values of 



(q^ + q^)d given by the last four equations. Also, introduce the 



1 2 l,2,i,4 



following notation: 



n = y (o) - iojc) S = smh q Jt 



s = sin q £ C = cosh q^l 



c = cos q I 

 Then Equations [39c,d] divided by n become: 



36 



