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

2017/08/03 by A. Yu. Anikin, Anikin, Anatoly, Dobrokhotov, Serguei +4
Earth and Planetary Sciences · Engineering · #Coastal and Marine Dynamics #FOS: Physical sciences #Mathematical Physics (math-ph) #Ocean Waves and Remote Sensing #Wave and Wind Energy Systems

paper · pdf · doi:10.48550/arxiv.1708.01107

openalex publication_date 2017/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the linear problem for water-waves created by sources on the\nbottom and the free surface in a 3-D basin having slowly varying profile\nz=-D(x). The fluid verifies Euler-Poisson equations. These (non-linear)\nequations have been given a Hamiltonian form by Zakharov, involving canonical\nvariables (\ξ(x,t),\η(x,t)) describing the dynamics of the free surface;\nvariables (\ξ,\η) are related by the free surface Dirichlet-to-Neumann\n(DtN) operator. For a single variable x\∈ bf R and constant depth, DtN\noperator was explicitely computed in terms of a convergent series. Here we\nneglect quadratic terms in Zakharov equations, and consider the linear response\nto a disturbance of D(x) harmonic in time when the wave-lenght is small\ncompared to the depth of the basin. We solve the Green function problem for a\nmatrix-valued DtN operator, at the bottom and the free-surface.\n

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