GROWING "WAVES DUE TO TRANSVERSE VELOCITIES 119 



+M (2.24) 



= 2 [^(y - 4m + lyo) - Ky - 4m - lyo)] 



m=— 00 



The extended definition of i/'2 (outside — /yo <y < ijo) is such that we may 



again take \pi = \p, , D% = DV at the ends of the interval. ?/i*DVi is 



still equal to — }4 at ij = ijo . Now 



+ 00 



£ [5(y - 4m + iW - ^(y — 4m - l^/o)] 



(2.25) 



= — 2 (—1)" sin /i„?/ 

 2/0 



so from (2.24) we may find 



v^L = -T (-l)"sin/xnj/ , ^ 



Matching to the external field as before gives 

 and applied to (2.26) we have 



00 /rfi 2 2\2 



_y^ = y (^ - uinn) , . 



The equations (2.23) and (2.27) for the even and odd modes may be 

 rewritten using the following reduced variables. 



. = ^« 



IT 

 1 _ Wj/0 _ Wo 



(2.23) becomes 



^' 4- 2 y ^ (n' - k^ _ _ . 



and (2.27) transforms to 



„^ 2^ + (n + 3^)2 [{n + 1^)2 - /c2]2 - s\{n + 3^)^ + k'] (2 99) 



= — tt;? 



