a reduced model for internal waves interacting with submarine ...

Notice that both approaches are fully dispersive regarding the bottom layer, since
the second coth is completely retained. For the shallow water upper layer regime
(kh 1 near zero) we obtain again the correct approximation of the full dispersion
relation, but here the termρ 1 k coth(kh 1 ) in the denominator is expanded with one
more term, namely
ρ 1
kh 1 coth(kh 1 )= ρ (
1
1+ (kh 1) 2
+ O ( (kh 1 ) 4)) ,
h 1 h 1 3
and consequently
ω 2 f=ω 2 h + O( (kh 1 ) 4) ,
while
ω 2 f=ω 2 r+ O ( (kh 1 ) 2) .
This means that the linear dispersion relation from the higher-order nonlinear
modelω 2 h is closer to the full (exact) linear dispersion relationω2 f
than the linear
dispersion relation from the lower-order modelω 2 r is. See Fig. 3.1 and a detail
in Fig. 3.2 where the corresponding phase velocities are depicted.
The inclusion of the higher order pressure term has improved the accuracy of
the phase speed over a much wider wavenumber band. This is very important in
reflection-transmission problems as shown by Muñoz and Nachbin in [23].
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