Asymptotics of Green function for the linear waves equations in a domain with a non-uniform bottom

Abstract : We consider the linear problem for water-waves created by sources on the bottom and the free surface in a 3-D basin having slowly varying profile $z=-D(x)$. The fluid verifies Euler-Poisson equations. These (non-linear) equations have been given a Hamiltonian form by Zakharov, involving canonical variables $(\xi(x,t),\eta(x,t))$ describing the dynamics of the free surface; variables $(\xi,\eta)$ are related by the free surface Dirichlet-to-Neumann (DtN) operator. For a single variable $x\in{\bf R}$ and constant depth, DtN operator was explicitely computed in terms of a convergent series. Here we neglect quadratic terms in Zakharov equations, and consider the linear response to a disturbance of $D(x)$ harmonic in time when the wave-lenght is small compared to the depth of the basin. We solve the Green function problem for a matrix-valued DtN operator, at the bottom and the free-surface.
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Anatoly Anikin, Serguei Dobrokhotov, Vladimir Nazaikinskii, Michel Rouleux. Asymptotics of Green function for the linear waves equations in a domain with a non-uniform bottom. Days on Diffraction 2017, Jun 2017, St. Petersbourg, Russia. pp.18-23, ⟨10.1109/DD.2017.8167988⟩. ⟨hal-01591657⟩

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