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

Appendix B
The Dirichlet-to-Neumann operator
Due to the importance of the Dirichlet-to-Neumann operator (DtN 0 ) in modelling
water waves we pointed here its relation with the Hilbert transform on the flat strip
used througout this work.
Given the problem
⎧
⎪⎨
⎪⎩
z υυ + z ωω = 0, −h≤ω≤0,
z ω = g(υ), ω=0,
z ω = 0, ω=−h,
the operatorT returns the tangential derivative z υ (υ, 0) from the Neumann data
z ω (υ, 0)=g(υ), that is
T [g]=z υ (υ, 0).
Therefore, its inverseT satisfies
g(υ)=z ω (υ, 0)=T [z υ (υ, 0)].
(B.1)
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