Internal Waves tn Channels of Vartable Depth 
bt » : : 
The boundary conditions are, if there is no free surface, 
ae (() era © Vo2(d) - (100) 
If we now multiply (91) by voz and (99) by vog, and integrate the 
resulting equations, by parts if necessary, andusing (92), (93), 
and (100) whenever possible, we obtain two equations the left-hand 
sides of which are identical. Taking the difference of these two 
equations, we have 
d | 
! = =. ot iB 
f | (s Pavao) ( Pot 8 P Aer I |» =0, (101) 
0 
which determines A,, since vy and vo are known, Then 
(99) can be integrated by the method of the variation of parameters 
to give vg» . Then V 29 is known. Further approximations follow > 
the same pattern. 
If the upper surface is free, the free-surface boundary 
condition can be found by integrating (99) in the Stieltjes sense, and 
an equation similar to (101) can be found. In fact, to obtain it one 
need only add the terms 
Zz 
= ! ON 
ff" p o(4) v5 9(4) + dW £8 (a) v 
2 0 00 ) 
2 
d)+4f d 
(4) d 98g (4) Myo 
to the left-hand side of (101). 
It remains to show that the f,(x) in (79) can be taken to 
be a constant. The argument is as follows. We have obtained succes- 
sive approximations to the eigenvalue and the eigenfunction, at each 
stage satisfying all the boundary conditions. If f(x) is not a constant, 
it is an additional term for the potential @¢ in (44), which gives rise 
to an additional velocity whose y component is zero, That velocity 
* Recall that a flat rigid upper surface is ruled out, and at the highest 
point of the symmetric channel x=0, _ so that V49 (4) does not have 
to vanish, 
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