Internal Waves tn Channels of Vartable Depth 
Ce eB ices ee gh , (106) 
where h is the average depth. Equation (106) agrees with the result 
of the classical shallow-water theory for long waves. Thus every 
comparison we have made indicates the correctness of our procedure. 
VIII. LONG WAVES 
We shall give some specific examples of the speeds of long 
internal waves, and shall consider two special classes of density 
stratification. 
(i) Exponential density distribution. If the density distribution 
is given by (61), we can work directly with (44), which in this case 
becomes (63), with \*% defined by (64). If we expand ¢ ina power 
series in k ? > we have 
2 4 
CA Pag. t Te Graygl¥ ht tids oP ayyl We detmnes 
(107) 
: 2 , : , 
Since we expect A to be negative for internal waves (i.e., waves 
that do not owe their existence primarily to the free surface), and 
since *% contains the factor o2 » which contains the factor k 2 
we shall write 
orion ty iF, tg 54.5 (108) 
in which 
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