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

Takingξ-derivatives,
φ ξ (ξ,ζ)= 1
2π
∞∑
k=−∞
k0
ĝ(k) i cosh( kπ
l
sinh ( kπ
l
( ))
ζ+
h 2
L
) e ikξπ/l .
The tangential derivative at the boundary is obtain makingζ→ 0:
φ ξ (ξ, 0)= 1
2π
∞∑
k=−∞
k0
ĝ(k) i cosh( kπ
l
sinh ( kπ
l
h 2
L
h 2
L
h 2
L
)
) e ikξπ/l .
The convergence is still uniform because of Eq. (C.4).
Therefore,
T [0,2l] [ f ](ξ)= 1
2π
∞∑
k=−∞
k0
( kπ
i coth
l
)
h 2
e ikξπ/l ˆf (k),
L
where
∫2π
ˆf (k)=
f (ξ)e −ikξ dξ, ξ=πξ/l,
0
that is, the Fourier coefficients inΠ[0, 2π].
It is also convenient to write the composition of one spatial derivative with
the Hilbert transformT [0,2l] [·] because they always come together in the models
considered here,
T [0,2l] [ f ] ξ (lξ/π)= 1
2π
∞∑
− kπ ( kπ
l coth l
k=−∞
k0
)
h 2
e ikξ ˆf (k).
L
Finally, for the discretization of the periodic domain (ignoring the aliasing
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