  We study Anderson localization of the classical lattice waves in a chain with
mass impurities distributed randomly through a power-law relation
$s^{-(1+\alpha)}$ with $ s $ as the distance between two successive impurities
and $\alpha>0$. This model of disorder is long-range correlated and is inspired
by the peculiar structure of the complex optical systems known as L\'evy
glasses. Using theoretical arguments and numerics, we show that in the regime
in which the average distance between impurities is finite with infinite
variance, the small-frequency behaviour of the localization length is $
\xi_\alpha(\omega) \sim \omega^{-\alpha} $. The physical interpretation of this
result is that, for small frequencies and long wavelengths, the waves feel an
effective disorder whose fluctuations are scale-dependent. Numerical
simulations show that an initially localized wavepacket attains, at large
times, a characteristic inverse power-law front with an $ \alpha $-dependent
exponent which can be estimated analytically.
