Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section,St. Petersburg Mathematical Journal

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Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section
St. Petersburg Mathematical Journal ( IF 0.8 ) Pub Date : 2020-09-03 , DOI: 10.1090/spmj/1626
S. A. Nazarov

Abstract:Cylindrical acoustic waveguides with constant cross-section $\omega$ are considered, specifically, a straight waveguide $\Omega ={\mathbb{R}}\times \omega \subset {\mathbb{R}}^d$ and a locally curved waveguide $\Omega ^\varepsilon$ that depends on a parameter $\varepsilon \in (0,1]$ . For

, in two different settings ( $\varepsilon =1$ and $\varepsilon \ll 1$ ), the task is to find an eigenvalue $\lambda ^\varepsilon$ that is embedded in the continuous spectrum $[0,+\infty )$ of the waveguide $\Omega ^\varepsilon$ and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator $\Delta +\lambda ^\varepsilon$ arises that vanishes at infinity and implies an eigenfunction in the Sobolev space $H^1(\Omega ^\varepsilon )$ . In the first case, it is assumed that the cross-section $\omega$ has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide $\Omega ^\varepsilon$ . In the second case, under an assumption on the shape of an asymmetric cross-section $\omega$ , the eigenvalue $\lambda ^\varepsilon$ is formed by scrupulous fitting of the curvature $O(\varepsilon )$ for small $\varepsilon >0$ .

中文翻译：

在具有恒定横截面的弯曲圆柱声波波导中捕获波

摘要： $\ omega$ 考虑了具有恒定横截面的圆柱声波导，具体而言，是取决于参数的直波导和局部弯曲波导。对于，在两个不同的设置（和）中，任务是找到一个固有值，该固有值嵌入在波导的连续光谱中，因此固有地不稳定。换句话说，出现了针对Helmholtz算子的Neumann问题的解，该解在无穷远处消失并暗示Sobolev空间中的本征函数。在第一种情况下，假定横截面 $\ Omega = {\ mathbb {R}} \ times \ omega \ subset {\ mathbb {R}} ^ d$ $\ Omega ^ \ varepsilon$ $\ varepsilon \ in（0,1]$

$\ varepsilon = 1$ $\ varepsilon \ ll 1$ $\ lambda ^ \ varepsilon$ $[0，+ \ infty）$ $\ Omega ^ \ varepsilon$ $\ Delta + \ lambda ^ \ varepsilon$ $H ^ 1（\ Omega ^ \ varepsilon）$ $\ omega$ 具有双对称性，并且对于波导轴的任何非平凡曲率都会产生特征值。在第二种情况下，在假设非对称横截面形状的情况下，特征值是通过对小的曲率进行严格拟合而形成的。 $\ Omega ^ \ varepsilon$ $\ omega$ $\ lambda ^ \ varepsilon$ $O（\ varepsilon）$ $\ varepsilon> 0$

更新日期：2020-09-10

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