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

where c 1 and c 2 are two functions of the wavenumber k. Using Fourier Series we
can write the general solution forη, at least in a formal manner, as
η(ξ, t)= 1
2π
∞∑
k=−∞
c 1 (k) exp ( iω(k)t ) e ikξ + 1
2π
∞∑
c 2 (k) exp ( −iω(k)t ) e ikξ .
k=−∞
Each term represents one wave mode, see [30]. Since the dispersion relationω(k)
is odd, each wave propagates in one direction: the first wave travels to the left, the
second one to the right. Returning to Fourier space, from the initial condition in
(4.11) we have,
Therefore,
and
⎧
c ⎪⎨ 1 (k)+c 2 (k)=̂η 0 (k),
⎪⎩ c 1 (k)−c 2 (k)= k
ω(k)û10(k).
(
c 1 (k)=0.5 ̂η 0 (k)+ k )
ω(k)û10(k)
(
c 2 (k)=0.5 ̂η 0 (k)− k )
ω(k)û10(k) .
(4.12)
For one propagation direction we set c 1 = 0, then
2c 1 =̂η 0 (k)+kû10(k)/ω(k)=0,
which implies the following relation between each amplitude of the initial condition
for u 1 and the amplitude of the initial condition forη:
û 10 (k)=− ω(k)
k ̂η 0(k), k0.
We use this relation to provide the initial condition for u 10 (by means of an FFT)
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