Growth of Resistance with Density of Scatterers in One Dimensional Wave Propagation

We study wave propagation in a one-dimensional disordered array of scattering potentials. We consider three different ensembles of scatterer configurations: an N-ensemble with a fixed number N of scatterers, an L-ensemble with a varying number of scatterers distributed over a fixed length L, and an NL-ensemble where both N and L are fixed. The latter ensemble allows a detailed study of the mean resistance and its variance for a fixed length L as the number of scatterers N increases. We find that the Landauer result, which predicts an exponential increase of the mean resistance with N, is valid only in the low-density regime. At high density the mean resistance grows exponentially with x / ~ and the concept of optical potential applies. In the crossover regime we find an interesting resonance.