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

and for similar reasons
η t +η x =− (u 1t + u 1x )− α 2 u 1(u 1t + u 1x )−
√ β
2
ρ 2 1
ρ 1 M(ξ) T[ M(˜ξ)(u 1tt + u 1xt ) ] ,
=− (u 1t + u 1x )+O ( α 2 ,α √ β,β ) . (2.36)
Substituting all these expressions in the evolution equation for u 1 we obtain
the evolution equation for the elevation of the interface,
η t +η x − 3 2 αηη x− ρ √
2 β 1
ρ 1 2 M(ξ) T[ ] (
M(˜ξ)η xt = O β,α 2 ,α √ β ) .
Finally, in curvilinear coordinates we have
η t + 1
M(ξ) η ξ− 3 2
α
M(ξ) ηη ξ− ρ √
2 β
ρ 1 2
1
M(ξ) T [η ξt]=0. (2.37)
This is an ILW equation with variable coefficients accounting for the slowly
varying bottom topography. Instead of the usual Hilbert transform on the halfspace,
a Hilbert transform on the strip appears. The dispersion relation for the flat
bottom case (M(ξ)=1) is
ω=
k
1+ ρ √
2 β
k coth( ). (2.38)
kh 2
ρ 1 2 L
The equation reduces to a regularized dispersive model in analogy with the Benjamin-
Bona-Mahony equation (BBM), [4].
Remarks:
1. The constant coefficient version of Eq. (2.37) differs from the ILW consid-
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