Internal Waves tn Channels of Vartable Depth 
where d is the maximum depth. Given 8 and m, ei is known, 
and (73) gives k. We canalso write (73) as 
kd 
o> = SEES I eebenntlaSt 16 ae (74) 
V2 V2 
: ne : ; 
With A given by (64), given 8B and k we canfind o from (74). 
Then \* is known and hence m?_ is known, 
Evidently there is also an antisymmetric mode, given by 
@= A sinh ax sinh yy cosk (z-ctt+ e) 
Then (69), (70), and (71) still stand but the dispersion formula is 
now 
oc = S— coth — ; (75) 
or 
coth see (76) 
V2- V2n 
It can be easily shown from (65) that if there is no free 
surface \* must be negative. The solutions given above all corres - 
pond to positive A 2 | Hence the presence of the free surface is es- 
sential for the existence of these solutions. The waves these solutions 
represent are therefore largely free-surface waves rather than in- 
ternal waves, and the density stratification has only a minor effect - 
that of affecting the value of the slope m. For this reason these so- 
lutions are not very interesting. We note that if B = 0 we have 
m* = 1, and (73) becomes 
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