Internal Waves tn Channels of Vartable Depth 
I o¢ ee ( for 5-0 
7 Ao t 8% 
rs) 
> ; (48) 
and we recognize that this is really the same as the equation of con- 
tinuity (31). We also recognize that (47) can be written in the much 
simpler form 
= OV. \. (47a) 
We now consider two wave numbers te and ear with the 
corresponding eigenvalues ax and cea , and the corresponding 
eigenfunctions $, and ¢, . The velocity components (u, » Vy > Wy ) 
and (uysVv, + Wy) satisfy 
Ou Ov ow 
1 1 bee 
x ts Oy i az pits (49) 
ou Ov ow, 
Se od cana it SOS ane ee) 
Multiplying (49) by ¢5 and integrating over the fluid domain, and 
utilizing the Green's Theorem and the boundary conditions (45) and 
(47a) for g=90, , b= 1 , V=V,, we have 
iE +k I + -@. AT Suna fs Le (51) 
in which 
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