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

assumption was made on the wave amplitude up to now, the model derived is
strongly nonlinear. It involves a Hilbert transform on the strip characterizing the
presence of harmonic functions (hence the potential flow) below the interface.
System (2.27) is a reduction of the original Euler equations constituted by
a pair of 2D-systems of PDEs to a single 1D-system of PDEs at the interface.
Instead of the integration of the Euler equations in the presence of a free interface,
a single 1D-system of PDEs is to be solved. Efficient computational methods can
be produced for this accurate reduced model which governs, to leading order, a
complex two-dimensional problem.
Remarks:
1. If the bottom is flat, M(ξ)=1 and the same system derived in [8] is recovered,
which is a nice consistency check.
2. When the lower depth tends to infinity (h 2 →∞) the limit for this model
is the same one obtained in [8] because the bottom is not seen anymore
(M(ξ)→1 and x(˜ξ, 0)→ ˜ξ). Therefore
φ xt (x, 0, t)→ 1 π
( ) ( )
(1−η)u1
xt ˜x, t
˜x− x
[ ((1−η)u1 )
]
d ˜x=H
xt
(x),
whereH is the usual Hilbert transform defined as
H[ f ](x)= 1 π
f ( ˜x)
˜x− x d ˜x.
In this (shallow upper layer) infinite lower layer regime, system (2.26) be-
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