Internal Waves tn Channels of Vartable Depth 
J 
oF [beelg Cee), ] + Blog ey) a, = 9, (10) 
at” 
in which the subscript L indicates the lower fluid and the subscript 
u the upper fluid and (¢,), = (¢ da ~ Py. If there is a free surface, 
Pp, = 0 there, and (10) reduces'to (5), With the form of ¢ (for any 
layer) given by (6), we can write (10) as 
0? [ (ef) y - (of), | = gl Reig es , (11) 
The differential system consists of (7) for each layer, (8), and (11) 
for each interface. 
Z Ze 
IV. VARIATION OF i OR c’ WITH k” FOR CONSTANT OR STEP- 
WISE DENSITY 
We can show that o* increases and c* decreasesas k# 
increases, ina very general manner, whether or not the fluid is stra- 
tified. Consider first the case p = constant, and let k? have two 
values, k? and k,?. The corresponding eigenvalues will be denoted 
by o,? and Se ,» and the corresponding eigenfunctions by f, and 
i, when 
2 
z Z 
) 
ee ei Oh are ee ees ; (12) 
1 b Wel 2 Z 
Le Sy. 
Va x= Me 2eSley Gilt, (13) 
and the free surface conditions are 
Jf, 
oy 
(14) 
1401 
