Internal Waves tn Channels of Vartable Depth 
bs 
3 BP'o 
Pou = aie ne ( Po a 3 ) vdy . (80) 
0 
Multiplying (45) by Pp, andusing (80), we have 
. bk EP 
1 3x Leo t 5 ) vdy + n, ew ee ; (81) 
0 
in which the boundary geometry not only determines n, and no, but 
will play a role after v has been differentiated with respect to x in 
the first term. We shall show later that the integral form of (81) can 
be changed to a differential form. 
If there is a free surface the condition there is (47) , which 
again must be expressed in terms of v. By virtue of (43), (47) can 
be written as 
a r gP ov. 
Applying the operator V on this and using (40), we have 
24 
Wed (3=—=1)/( 
4 
Equations (78), (81), and (82) constitute the differential system 
defining the eigenvalue problem, 
1417 
