Internal Waves tn Channels of Vartable Depth 
If the inertial effect of the density variation is neglected (Boussinesq 
approximation), this equation becomes 
+ oN Pie Oy 4 (115) 
1 a ! 
00 3 * 0 00 
the solution of which satisfying the boundary condition at y =0 is 
Jory), where Jo is the Bessel function of the zeroth order, 
If the upper surface is free, then (47) gives the condition 
=- = ! 
Bey (4) = o'y5(€) » (116) 
so that (116) is replaced by 
BJ (vd) = yJ)(vd) . (47) 
0 
which gives Y. Once y is known, the long-wave speed cg is calculat- 
ed from 
) Ss ciskadt OP4 ce Aces cee (118) 
It is important to note that the roots of (116) or of (117) are 
for internal waves only. The speed of waves due predominantly to the 
presence of the free surface is found in the following way. First of 
all, differentiation of (46) with respect to t gives directly 
2 
BPy7 ® 
i ee (119) 
op +g! 
08? 9 
We see that there are no terms free of k in (63) and (119). Hence 
any solution of (114), which automatically satisfies condition (45) at 
the channel boundary, is an acceptable solution. But at this stage we 
cannot determine cy). Proceeding to the second approximation, whe- 
ther or not the Boussinesq approximation is used, we reach a nonho- 
mogeneous differential equation in ?>(y) , the solution of which to- 
gether with the boundary conditions then determines Yor c,. The Co 
so determined is not proportional to VB, but is much larger, and the 
corresponding waves are predominantly surface waves, the density 
stratification merely causing a minor correction if B is small. 
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