Internal Waves tn Channels of Vartable Depth 
which could have been obtained by integrating (91) across the free 
surface in the Stieltjes sense. 
If f(y) is a constant, (91) becomes 
Pg Sigg rie i hgee Brad he 
000 
which agrees with the equation governing wave motion (Yih 1965, 
p. 29) ina stratified fluid for k=0. Furthermore, (95) becomes 
Which is the free-surface boundary condition for a rectangular chan- 
nel, and which agrees with (82) when v is independent of x and 
when k = 0 (for which \ = Xo). The conditions (92) and (93) remain 
unchanged if f(y) is constant. Hence the differential system consist- 
ing of (91), (92) and (93), or (91), (92) and (95), agrees, as it 
should, with that for a rectangular channel when f(y) is taken to be a 
constant. 
From the system (91), (92) and (93) or (91), (92) and 
(95) we candetermine Ag and vgo(y). Then vgo(y) is known from 
(90). 
We shall describe the next stage of approximation, The pro- 
cedure of successive paaroximation will then be clear. Taking terms 
of order k2 in (78), we have 
2 ! 1 
> i ss e 
ity oro pegepnOk MABE “20)| on 
1421 
