NOs 
k,6 k,6 
x) 77a ,e sin(ka - wa + «) -a,e sin(k,a - wot + €5) 
k, 6 k,6 
(24) Zz) =a,e cos(kja-w t+e))+a,e cos(k,a - wt + €,) 
Pp, = 0 
ao) k,6 AW k,6 
Fy - K e sin(k,a - wt + €)+ iS e sin(k,a- watt t,) 
It is irrotational to first order. For simplicity, we assume that w5 > w)- 
The second order equations to be solved are obtained by substitut - 
ing (24) into (7) where y is set equal to $8 only and all partials with 
respect to B vanish. These equations are given by 
oeeuec2q ibaqe! Pls 0 
3 2 2ks 2k, 6 
224 * 8225 + Pog/p = gla, ke ~ +a, ky e 
(k, +k,)6 
+ 2a,ajk)k,e cos(k, - k))a-(w,-w)t te, -€,)) 
(k,+ k,)6 
[25] [x5 +25 -2a)a,e cos((k, -k,)a-(w,-w,)t+ €5-&) }, = 0 
(AG PANS t6) 2k,6 
= Zz eal 1 2 2: 
SE anlot aoe er ety 
(k) +k, )6 
+ aja,(wk, + w>k)) e cos((k, - k))a- (o, - w,)t + €& -€) ) Jda 
aa (k,+k )6 
Z (w, - @1)(w5%)) e sin((k, - k,)a- (w, - w,)t + €- €,) ]aé 
+ [lzoe7 
In (25), we have four equations inp, z and x. The first three de- 
termine a solution, subject to appropriate boundary conditions, for p, z, 
and x. The last is then used to see if the solution is irrotational and to 
make the solution irrotational if possible. Note that x = x, (6)t can be 
added to (23), if desired, as part of the total solution to the equations. The 
boundary conditions are that Dai O at 6 = 0 and that X> and Z5 approach 
zero as 6 approaches - © 
If the right hand side of the second equation is represented by 
g[F(5) + G(a, 6,t)] and the third equation is written as x, + Zo 6 -G(a, 6, t) =0, 
2a 
