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

Apply the Fourier Transform to problem (C.1):
⎧
⎪⎨
⎪⎩
(−ik) 2 ( π
l)2
ˆφ+ ˆφζζ = 0, on−h 2 /L≤ζ≤ 0,
ˆ
φ ζ (k, 0)=ĝ(k),
ˆ
φ ζ (k,−h 2 /L)=0.
(C.3)
The solution for problem (C.3) is
φ(k,ζ)=
ˆ ĝ(k) cosh ( kπ
l
πk/l
Therefore, for C 1 initial Neumann data g,
φ(ξ,ζ)= 1
2π
= 1
2π
∞∑
k=−∞
∞∑
k=−∞
k0
sinh ( kπ
l
ˆ
φ(k,ζ)e ikξ ,
ĝ(k) cosh ( kπ
l
πk/l
( ))
ζ+
h 2
L
) , k0.
h 2
L
sinh ( kπ
l
( ))
ζ+
h 2
L
) e ikξ +
h 2
L
ˆ
φ(0)
2π ,
the convergence is uniform in 0≤ξ≤2π and also in−h 2 /L≤ζ≤ 0 since
∣
cosh ( kπ
l
for all integer k0.
sinh ( kπ
l
( ))
ζ+
h 2
∣ ∣∣∣∣∣∣∣
L
)
∣ ≤ cosh ( kπ
l
sinh ( kπ
We wantφ ξ ; returning to the original variables
φ(ξ,ζ)= 1
2π
h 2
L
∞∑
k=−∞
k0
ĝ(k) cosh ( kπ
l
πk/l
l
h 2
L
h 2
L
sinh ( kπ
l
)
∣ ∣∣∣∣∣
)
π coth( ∣ = l
( ))
ζ+
h 2
L
) e ikξπ/l +
h 2
L
)∣
h ∣∣∣∣∣
2
, (C.4)
L
ˆ
φ(0)
2π .
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