Chta-Shun Yth 
Because of the convenience afforded by the exponential density 
stratification, we have used the differential equation in @ instead of 
(78). Remembering (61), (86), 
and 7) 
" l RED f 
0 aad a Ge 
one can readily show that (114) is equivalent to (91). In fact, ' poly) 
is proportional to Pon ‘ 
(ii) Superposed homogeneous layers. If the fluid is composed of 
superposed layers, in each layer the governing equation is (2) or (7), 
The boundary conditions are (8) for the rigid channel boundary, (11) 
for the interfaces, and (9) for the free surface. Of course, (9) isa 
special case of (11). Note that the f in (7) is not the f in (83). 
The solution is now not restricted to symmetric channels. 
Suppose there are n layers. We shal use the expansion 
fa ¥) => f tks tc Kk of FP Sees (120) 
for the mth layern counting from the bottom up. Furthermore, we 
shall write 
o = 0 +kao +ko amecweate (121) 
Substituting (120) and (121) into (7) and taking only terms free from 
k, we see that the solution is 
f ek oi reece (122) 
which satisfy all the boundary conditions if only terms free from k 
are taken, 
For the second approximation we have to solve the equations 
— + — = — 
( 7) 5) f 2 Ca for? mle why2,.o.emi ib ae 
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