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

Again, ω2
k 2
large.
→ 0 as k→∞, so bounded phase velocities are obtained as k becomes
Now, let us make a comparison between the different dispersion relations obtained
throughout this work. Initially, we have the dimensional full dispersion
relation
ω 2 f=
g(ρ 2 −ρ 1 )k 2
ρ 1 k coth(kh 1 )+ρ 2 k coth(kh 2 )
(3.11)
that comes from the linearized Euler equations around the undisturbed state, see
for example [18]. To compare it with the dimensionless dispersion relations (2.29)
and (3.10) it must be taken into account that because of the non-dimensionalization,
k= k L , ω= U 0
L ω.
Therefore, (2.29) in dimensional form becomes
ω 2 g(ρ 2 −ρ 1 )k 2
r= ρ 1
h 1
+ρ 2 k coth(kh 2 ) . (3.12)
As it has been stated, the reduced model (2.27) captures the dispersion relation for
the shallow water (long waves) regime in the upper layer since
ρ 1 k coth(kh 1 )→ ρ 1
h 1
,
as kh 1 → 0.
On the other side, the relation (3.10) in dimensional form is
ω 2 h = g(ρ 2 −ρ 1 )k 2
ρ 1
h 1
+ 1h 3 1ρ 1 k 2 +ρ 2 k coth(kh 2 ) .
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