2016/09/30 by Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, С. А. Назаров +1 · 20 citations
Computer Science · Engineering · Mathematics · #Acoustics #Advanced Mathematical Modeling in Engineering #Computer science #Mathematical analysis #Mathematics #Numerical methods in inverse problems #Optics #Physics #Plane wave #Reflection (computer programming) #Sound transmission class #Stability and Controllability of Differential Equations #Upper and lower bounds #Wavenumber #math.AP #msc:35J05 #msc:35R30 #msc:65N21 #msc:65N25 #msc:78A40 #msc:78A46
paper · pdf · doi:10.1016/j.matpur.2017.07.020
published in Journal de Mathématiques Pures et Appliquées 111, 79-105 (Elsevier BV) · Journal de Mathématiques Pures et Appliquées, 12/08/2017
openalex publication_date 2017/08/12 · arxiv created 2017/09/18 · arxiv updated 2017/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.