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We consider the propagation of acoustic waves at a given wavenumber in awaveguide which is unbounded in one direction. We explain how to constructpenetrable obstacles characterized by a physical coefficient $\rho$ which areinvisible in various ways. In particular, we focus our attention oninvisibility in reflection (the reflection matrix is zero), invisibility inreflection and transmission (the scattering matrix is the same as if there wereno obstacle) and relative invisibility (two different obstacles have the samescattering matrix). To study these problems, we use a continuation method whichrequires to compute the scattering matrix $\mathbb{S}(\rho)$ as well as itsdifferential with respect to the material index $d\mathbb{S}(\rho)$. Thejustification of the method also needs for the proof of abstract results ofontoness of well-chosen functionals constructed from the terms of$d\mathbb{S}(\rho)$. We provide a complete proof of the results in monomoderegime when the wavenumber is such that only one mode can propagate. And wegive all the ingredients to implement the method in multimode regime. We endthe article by presenting numerical results to illustrate the analysis.

A continuation method for building invisible obstacles in waveguides