Perfect transmission invisibility for waveguides with sound hard walls

被引:19
作者
Bonnet-Ben Dhia, Anne-Sophie [1 ]
Chesnel, Lucas [2 ]
Nazarov, Sergei A. [3 ,4 ,5 ]
机构
[1] Univ Paris Saclay, CNRS, Ensta Paris Tech, Lab Poems,ENSTA,INRIA, 828 Blvd Marechaux, F-91762 Palaiseau, France
[2] Univ Paris Saclay, INRIA, Ecole Polytech, Ctr Math Appl, Route Saclay, F-91128 Palaiseau, France
[3] St Petersburg State Univ, Univ Skaya Naberezhnaya 7-9, St Petersburg 199034, Russia
[4] Peter Great St Petersburg Polytech Univ, Polytekh Skaya Ul 29, St Petersburg 195251, Russia
[5] Inst Problems Mech Engn, Bolshoy Prospekt 61, St Petersburg 199178, Russia
来源
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES | 2018年 / 111卷
基金
俄罗斯基础研究基金会;
关键词
Invisibility; Acoustic waveguide; Asymptotic analysis; Scattering matrix; CONTINUOUS-SPECTRUM; ASYMPTOTIC EXPANSIONS; ENFORCED STABILITY; SCATTERING; EIGENVALUE; MEDIA; PROPAGATION; FREQUENCIES; THRESHOLDS; MODEL;
D O I
10.1016/j.matpur.2017.07.020
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We are interested in a time harmonic acoustic problem in a waveguide with locally perturbed sound hard walls. We consider a setting where an observer generates incident plane waves at -infinity and probes the resulting scattered field at -infinity and +infinity. Practically, this is equivalent to measure the reflection and transmission coefficients respectively denoted R and T. In [9], a technique has been proposed to construct waveguides with smooth walls such that R = 0 and vertical bar T vertical bar = 1 (non-reflection). However the approach fails to ensure T =1 (perfect transmission without phase shift). In this work, first we establish a result explaining this observation. More precisely, we prove that for wavenumbers smaller than a given bound depending on the geometry, we cannot have T = 1 so that the observer can detect the presence of the defect if he/she is able to measure the phase at +infinity. In particular, if the perturbation is smooth and small (in amplitude and in width), k(*). is very close to the threshold wavenumber. Then, in a second step, we change the point of view and, for a given wavenumber, working with singular perturbations of the domain, we show how to obtain T = 1. In this case, the scattered field is exponentially decaying both at -infinity and +infinity. We implement numerically the method to provide examples of such undetectable defects. (C) 2017 Elsevier Masson SAS. All rights reserved.
引用
收藏
页码:79 / 105
页数:27
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