Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section

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 $d>2$, 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 $\varepsilon >0$.