  The time that waves spend inside 1D random media with the possibility of
performing L\'evy walks is experimentally and theoretically studied. The
dynamics of quantum and classical wave diffusion has been investigated in
canonical disordered systems via the delay time. We show that a wide class of
disorder--L\'evy disorder--leads to strong random fluctuations of the delay
time; nevertheless, some statistical properties such as the tail of the
distribution and the average of the delay time are insensitive to L\'evy walks.
Our results reveal a universal character of wave propagation that goes beyond
standard Brownian wave-diffusion.
