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

Appendix A
Approximation for the horizontal
derivatives at the unperturbed
interface
We want to justify the use of
√ β
M(ξ) T[ M(˜ξ)u 1tx
]
instead of
( √ β
M(ξ) T[ M(˜ξ)u 1t
])x in
the substitution ofη x in system (2.30), in the case of slowly varying topography.
Hence we identify 1
M(ξ) T[ M(˜ξ)u 1t
]
as the tangential derivative of the solution of
the Laplace equation with Neumann conditions defined in the auxiliary problem
⎧
⎪⎨
⎪⎩
Φ xx +Φ zz = 0, on− h 2
L + h 2
L h(εx)≤z≤0,
Φ z = u 1t ,
at z=0,
Φ z −ε h 2
L h′ (εx)Φ x = 0, at z=− h 2
L + h 2
L h(εx),
(A.1)
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