Perfect transmission invisibility for waveguides with sound hard walls

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.