a reduced model for internal waves interacting with submarine ...
a reduced model for internal waves interacting with submarine ...
a reduced model for internal waves interacting with submarine ...
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effect) we have that<br />
T [0,2l] [ f ] ξ (ξ j )≈ 1<br />
2π<br />
∑N/2<br />
k=−N/2+1<br />
k0<br />
− kπ l coth ( kπ<br />
l<br />
)<br />
h 2<br />
e ikξ j<br />
ˆf (k), j=1,..., N,<br />
L<br />
where<br />
N∑<br />
ˆf (k)=∆ξ f (ξ j )e −ikξ j<br />
.<br />
j=1<br />
In conclusion, the symbol of the operatorT [0,2l] [·] ξ (that is the composition of<br />
one spatial derivative <strong>with</strong> the Hilbert trans<strong>for</strong>m) in the new coordinateξ is<br />
− kπ l coth ( kπ<br />
l<br />
)<br />
h 2<br />
.<br />
L<br />
97
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