Chta-Shun Yth 
therefore cannot possibly satisfy (45) unless its x-component is 
zero - or unless £ (x) is a constant. 
In the next section we shall study long waves. But before 
we leave this section we shall make two comments: (i) The method 
of expansion can be slightly modified to deal with unsymmetric chan- 
nels. (ii) If the fluid is not stratified, Po is constant, and from 
(90) one deduces that vgg vanishes and with it v290 2lso vanishes, 
according to (88). However, for the second approximation, in which 
terms of 0(k? ) are considered, it is found that Vo2 is governed by 
the equation 
=O; (102) 
which is what (99) becomes. The boundary condition on vo9 is 
identical with (95) : 
u f'(d) 
V" 92 (4) = [ays - oa | V2 (4) , (103) 
! ee a + 
mado heey yea cis (104) 
and 
4 fy) 
72 105 
f 
0 
Substitution of (104) and (105) into (103) gives 
1424 
