TY - JOUR

T1 - Perfect transmission invisibility for waveguides with sound hard walls

AU - Bonnet-Ben Dhia, A.-S.

AU - Chesnel, L.

AU - Nazarov, S.A.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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 −∞ and probes the resulting scattered field at −∞ and +∞. 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 |T|=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 k ⋆ 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 +∞. 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 −∞ and +∞. We implement numerically the method to provide examples of such undetectable defects.

AB - 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 −∞ and probes the resulting scattered field at −∞ and +∞. 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 |T|=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 k ⋆ 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 +∞. 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 −∞ and +∞. We implement numerically the method to provide examples of such undetectable defects.

KW - Acoustic waveguide

KW - Asymptotic analysis

KW - Invisibility

KW - Scattering matrix

UR - http://www.scopus.com/inward/record.url?scp=85028304927&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2017.07.020

DO - 10.1016/j.matpur.2017.07.020

M3 - Article

VL - 111

SP - 79

EP - 105

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 3

ER -