TY - JOUR

T1 - Surface waves in a channel with thin tunnels and wells at the bottom

T2 - Non-reflecting underwater topography

AU - Chesnel, Lucas

AU - Nazarov, Sergei A.

AU - Taskinen, Jari

PY - 2020/1/1

Y1 - 2020/1/1

N2 - We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.

AB - We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.

KW - asymptotic analysis

KW - invisibility

KW - Linear water-wave problem

KW - scattering matrix

KW - weighted spaces with detached asymptotics

UR - http://www.scopus.com/inward/record.url?scp=85087825410&partnerID=8YFLogxK

U2 - 10.3233/ASY-191556

DO - 10.3233/ASY-191556

M3 - Article

AN - SCOPUS:85087825410

VL - 118

SP - 81

EP - 122

JO - Asymptotic Analysis

JF - Asymptotic Analysis

SN - 0921-7134

IS - 1-2

ER -