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

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 (ξ(x, t), η(x, t)) describing the dynamics of the free surface; variables (ξ, η) are related by the free surface Dirichlet-to-Neumann (DtN) operator. For a single variable x ϵ R and constant depth, DtN operator was explicitly 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 basin's depth. We solve the Green function problem for a matrix-valued DtN operator, at the bottom and the free-surface.