具有声硬壁的波导的完美传输不可见性
Perfect transmission invisibility for waveguides with sound hard walls
摘要：
我们感兴趣的是具有局部扰动声硬壁的波导中的时谐声学问题。我们考虑这样一种设置，其中观察者在$-\infty$处产生入射平面波，并在$-\infty$和$+\infty$处探测所产生的散射场。实际上，这相当于测量分别表示为$R$和$T$的反射和透射系数。在文献[9]中，提出了一种技术来构造具有光滑壁的波导，使得$R=0$和$|T|=1$（无反射）。然而，该方法不能确保$T=1$（无相移的完美传输）。在这项工作中，我们首先建立一个解释这一观察结果的结果。更确切地说，我们证明，对于小于给定的约束$k_{\star}$取决于几何形状的波数，我们不能有$T=1$，使观察者可以检测到缺陷的存在，如果他/她能够测量相位在$+\infty$。特别地，如果扰动是平滑且小的（在幅度和宽度上），则$k_{\star}$非常接近阈值波数。然后，在第二步中，我们改变的观点，对于一个给定的波数，工作与奇异扰动的域，我们展示了如何获得$T=1$。在这种情况下，散射场在$-\infty$和$+\infty$处均呈指数衰减。我们实现数值的方法，提供这样的不可检测的缺陷的例子。
英文摘要：
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 $-\infty$ and probes the resulting scattered field at $-\infty$ and $+\infty$. 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 $k_{\star}$ 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 $+\infty$. In particular, if the perturbation is smooth and small (in amplitude and in width), $k_{\star}$ 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 $-\infty$ and $+\infty$. We implement numerically the method to provide examples of such undetectable defects.
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