Accurate evaluation of the effects of random disorder on the vibration localization in chains of damped oscillators

The focus of this paper is on a reassessment of the quantification of the vibration localization induced by random disorder in chains of damped oscillators. It is in particular shown that the standard measure of localization, the mean localization factor, is often strongly dependent on damping, not on disorder, and thus does not represent an appropriate characterization of its effects. Next, it is found that the variance of the localization factor is much more sensitive to disorder and thus is a better measure of the effects of disorder on vibration localization. Asymptotic approximations of the mean and variance were then obtained for the following limits: (i) R >> σ and R >> δ, (ii) R << and R << δ, and (iii) R >> σ, δ arbitrary where R, σ, and δ are dimensionless measures of the coupling between oscillators, disorder, and damping. To complement these quantitative results, a qualitative discussion of the transition zone R ≈ was performed. Interestingly, it is found that this variance does exhibit a peak at an intermediate coupling level when a damping parameter exceeds the standard deviation of the disorder.