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

problem (2.22). Therefore,
P x =− ρ (
2
η x + √ 1
β
ρ 1 M(ξ) T[ (
M(˜ξ)η tt x(˜ξ, 0), t ) ]
)
(ξ) + O(β). (2.25)
Now,φ x (x, 0, t) is a tangential derivative on the flat upper boundary for problem
(2.22), whose domain is a corrugated strip. Hence, it is also expressed as a
Hilbert transform acting on Neumann data. Since
φ x (x, 0, t)=
L
2h 2 M ( ξ(x, 0) )
(
M(˜ξ)η t x(˜ξ, 0), t ) ( πL
coth
(˜ξ−ξ(x, 0) )) d ˜ξ,
2h 2
a Hilbert-like transform on the corrugated strip has been identified as:
T c [ f ](x)=
L
2h 2 M ( ξ(x, 0) )
M(˜ξ) f ( x(˜ξ, 0) ) ( πL
coth
(˜ξ−ξ(x, 0) )) d ˜ξ,
2h 2
which is not a convolution operator, unlike Eq. (2.24).
Finally, substituting the expression for P x obtained in Eq. (2.25) in the upper
layer averaged equations (2.18) gives
⎧
η t − [ (1−η)u 1
]x
(
= 0,
⎪⎨ u 1t + u 1 u 1x + 1− ρ )
2
η x =
ρ 1
√
L ρ 2 1
⎪⎩ β
2h 2 ρ 1 M ( ξ(x, 0) ) (
M(˜ξ)η tt x(˜ξ, 0), t ) ( πL
coth
(˜ξ−ξ(x, 0) )) d ˜ξ+ O(β).
2h 2
In a compact notation this becomes
⎧
η
⎪⎨ t − [ (1−η)u 1
]x
(
= 0,
⎪⎩ u 1t + u 1 u 1x + 1− ρ )
2
η x = √ β ρ 2 1
ρ 1 ρ 1 M(ξ) T[ ( )]
M(·)η tt x(·, 0), t (ξ)+O(β),
24