Vibration and wave localization in a nearly periodic beaded string

The dynamics of an exact, linear model of a string with attached beads are examined. A study of the perfectly periodic system is performed using a transfer matrix formulation. A closed-form solution for the natural frequencies of the finite system is obtained. The relationship between natural frequencies and the passbands and stop bands of propagating waves is studied. The effect of random disorder of (a) bead spacing and (b) bead mass is examined and interesting, fundamental differences are observed. In the case of bead-spacing disorder both weak and strong localization occur, whereas bead-mass disorder only causes weak localization. Localization, the spatial confinement of vibration energy due to periodicity breaking disorder, is quantified by the localization factor. Analytical approximations of the localization factor are derived, in the limits of large and small coupling to disorder ratios. The results are verified through Monte Carlo simulations. The natural frequencies and modes of the disordered system are examined. A dispersion of the natural frequencies is evidenced, presented as a curve veering phenomenon. The existence of localized mode shapes is demonstrated.