A continuation method for building invisible obstacles in waveguides

We consider the propagation of acoustic waves in a waveguide which is unbounded in one direction. We explain how to construct at a given wavenumber penetrable obstacles characterised by a physical coefficient $\rho$ which are invisible in various ways. In particular, we focus our attention on invisibility in reflection (the reflection matrix is zero), invisibility in reflection and transmission (the scattering matrix is the same as if there were no obstacle) and relative invisibility (two different obstacles have the same scattering matrix). To study these problems, we use a continuation method which requires to compute the scattering matrix $\mathbb{S}(\rho)$ as well as its differential with respect to the material index $d\mathbb{S}(\rho)$. The justification of the method also needs for the proof of abstract results of surjectivity of well-chosen functionals constructed from the terms of $d\mathbb{S}(\rho)$. We provide a complete proof of the results in monomode regime when the wavenumber is such that only one mode can propagate. And we give all the ingredients to implement the method in multimode regime. We end the article by presenting numerical results to illustrate the analysis.