The arbitrary functions of y can be evaluated by means 

 of the boundary conditions U. = V^ = on x = OjT. VJe have 



2/3 o 2 ? 

 sin nsy C^ - ^~ q- [(nsy^ + en s )sin nsy + y cos nsy] (^6) 



' -> O 



sin nsy A-j_ = i_— . [(nsy + 9n s )sin nsy + y cos nsy] iSl) 



C r: O-^o^JlI [2nsy sin nsy + (y^n^s^ + 2 + 9n3s3) • 



• cos nsy][.^ - ve^^^ + e^^^] - e^~^^(^ + i)cos nsy j- (^-8) 



sin nsy C = _X-. 4 [ 2£^/3(y^ns + .L + Qn^s^)irE^''^ - r^ - gS/B) + 

 ^ \/3 Q L ns 



2e ?9e^^^n • / c- k . > 2/3 



+ ^ _ ^^-£- — ]sin nsy + ( 5 y cos nsy — -i- sm nsy)re - 



3 3 ^2 



- (9y cos nsy - ^ sin nsy)e +(re -e) (y^ns+Qn^s )sin nsy k 



ns J 



('^9) 

 The first-order contribution to H can be found from 

 equations (12), (I3). The first order equations are 



9U 3H- 



ns 

 5t 



2 - nsy V, + Q — i = nseAU^ 



■^ 1 ax 1 



av_ aH. 



ns —-i^ + nsy U-. + Q —-i - nseAV, 



at; J- av 1 



from which H-, is found to be 



H = - £i5_£.££Jl-^[(Qns + y^)cos nsy + (y^ns + y9n^s^)sin nsy] * 

 9^ 



• rl(x2 + r^) + (e^^3 - r)(x + e^^^) ] + -©. cos nsy + 

 ^ 2 ns 



g2/3^_9ns ^ 3_wy sin nsy _^ cos nsy^ L 

 3 ns n2s2 



