  Structures with heavy-tailed distributions of disorder occur widely in
nature. The evolution of such systems, as in foraging for food or the
occurrence of earthquakes is generally analyzed in terms of an incoherent
series of events. But the study of wave propagation or lasing in such systems
requires the consideration of coherent scattering. We consider the distribution
of wave energy inside 1D random media in which the spacing between scatterers
follow a L\'evy $\alpha$-stable distribution characterized by a power-law decay
with exponent $\alpha$. We show that the averages of the intensity and
logarithmic intensity are given in terms of the average of the logarithm of
transmission and the depth into the sample raised to the power $\alpha$.
Mapping the depth into the sample to the number of scattering elements yields
intensity statistics that are identical to those found for Anderson
localization in standard random media. This allows for the separation for the
impacts of disorder distribution and wave coherence in random media.
