igs 
The full solution is obtained by combining equations (5), (24), 
and (27). This solution satisfies the equations to second order. It is 
irrotational to second order. 
If more terms are added to the linear solution subject to the con- 
dition that w) < W5 < Ww, Kseed wy the terms in the linear solution inter - 
act in a predictable way to generate appropriate second order terms. 
The randomized second order solution for x(a,6,t) and z(a,6,t) are 
given by equations (28). 
k.§ 
x(alont)="ae—= aa. e 7 sin(k.a-w.t + e.) 
i i i i 
a.a, eae (k.+k.)6 
= =z z Talc e J 1 abet (gs Sho ae SG) 
jie 
jrit g a 
a.a, (k.-k.)6 
+ EEA. tw.)w,e J * gin((k.-k,)a-(w.-w,)t+e,-€.) 
jpii 8s jo Jot Joi Joi 
2k.6 
+Zavu.k.e *t 
(28) 
k,6 
ZA(,O5 6) =) Oates e cos(k.a-w.t +.«€.) 
i i i i 
aia. 9 2 spe 
+ # nares (w; PEEP APs Je eos ( (ess i) aio aa ec een) 
aa. (k,-k,)6 
= pa Oe —J w (w.+w.)e J cos((k.-k.)a-(w.-w.)t+ €.-€.) 
j>i i Sd a) eee J ekesl Ay et Jou 
The solution given by equation (28) has features that are quite 
different from solutions obtained by the assumption of irrotationality in 
the Eulerian system. The trigonometric second order terms involve only 
the difference between two frequencies. One of the second order terms 
dies out rapidly with depth. The other dies out slowly with depth. The 
term that is linear in t is the average drift of the particle, and as a group 
of higher waves passes, the effect of the second order terms is to increase 
the drift. Correspondingly, as low waves pass, the drift is decreased. 
The positions of those particles that are on the free surface are 
obtained by setting 6 = 0, and the result is equation (29). 
