Scattering and embedded trapped modes for an infinite nonhomogeneous Timoshenko beam

We consider an infinite Timoshenko beam whose density is weakly perturbed on a finite interval. The plane waves of the unperturbed system form the continuous spectrum, which has multiplicity 2 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0 < \omega^2 < \omega^2_0}$$\end{document} and multiplicity 4 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega^2 > \omega^2_0}$$\end{document} (here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} is the frequency and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega_0}$$\end{document} is the cut-off frequency). The first branch of the spectrum (the flexural mode) corresponds to the range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega^2 >0 }$$\end{document}, and the second (the shear mode) to the range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega^2>\omega_0^2}$$\end{document}. The perturbation gives rise to a resonance (a pole of the analytic continuation of the reflection coefficient of the corresponding scattering problem) in a neighborhood of the point β = 0, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta^2=\omega_0^2-\omega^2}$$\end{document}, in the complex plane of the parameter β. When the pole is real and positive, it becomes an eigenvalue of the problem. It turns out that this is the case for certain perturbations (e.g., a square barrier of certain length); the pole then defines an eigenvalue (a trapped mode) embedded in the continuous spectrum of the first branch. When the conditions for the existence of an eigenvalue are not satisfied, the reflection coefficient changes abruptly in a neighborhood of the real part of the pole, rapidly growing from almost zero to almost one and then back to zero, as β passes through the resonant value, according to a formula of Breit–Wigner type familiar from quantum mechanics.