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.

中文翻译：

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在波导中建立看不见的障碍的一种连续方法

我们考虑声波在一个方向上不受限制的波导中的传播。我们解释了如何在给定的波数下构造以物理系数| $ \ rho $ |为特征的可穿透障碍。它们以各种方式不可见。特别是，我们将注意力集中在反射的不可见性（反射矩阵为零），反射和透射的不可见性（散射矩阵与没有障碍物相同）和相对不可见性（两个不同的障碍物具有相同的散射矩阵） ）。为了研究这些问题，我们使用一种连续方法，该方法需要计算散射矩阵| $ \ mathbb {S}（\ rho）$ | 以及与物料索引| $ d \ mathbb {S}（\ rho）$ |⁠的差异。该方法的合理性还需要证明根据| $ d \ mathbb {S}（\ rho）$ |⁠项构造的精心选择的泛函的抽象性的抽象结果。当波数使得仅一种模式可以传播时，我们提供了单模态结果的完整证明。并且我们给出了在多模式方案中实施该方法的所有要素。我们通过提供数值结果来说明分析来结束本文。