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

Setξ=πξ/l. In the new coordinatesφ(ξ,ζ)=φ(ξ,ζ) satisfies
⎧
) 2φξξ
+φ ζζ = 0, −h 2 /L≤ζ≤ 0, 0≤ξ≤2π,
( π
2
⎪⎨
⎪⎩
φ(0,ζ)=φ(2π,ζ),
φ ζ (ξ, 0)=g(ξ),
φ ζ (ξ,−h 2 /L)=0.
(C.1)
We consider the Fourier Series inξ∈[0, 2π] with its coefficients given by
ˆ f (k)=
∫ 2π
0
f (ξ)= 1
2π
f (ξ)e −ikξ dξ,
∞∑
k=−∞
ˆ f (k)e ikξ .
(C.2)
The Discrete Fourier Transform (DFT) inξ is
ˆ f (k)=∆ξ
N∑
j=1
f (ξ j )e −ikξ j
, ξ j = j∆ξ, ∆ξ= 2π
N ,
exactly the same one used by Trefethen [28] together with the inverse
f (ξ j )= 1
2π
∑N/2
k=−N/2+1
ˆ f (k)e ikξ j
, j=1,..., N,
where k∈{−N/2+1,..., N/2} because in the discrete domain e ik j∆ξ = e ik j2π/N so
there is no difference for k=k 0 mod N.
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