The Statistics of Waves Propagating in a One-Dimensional Random Medium

We consider the one-dimensional scattering of waves in a time-independent random medium. The waves considered are time-harmonic. It is assumed that the wavelength of the waves and the correlation length of the scatterers are small compared with the distance required for significant scattering. Stochastic process theory is used to investigate the statistics of the wavefield. The problem of a wave incident on a length of random medium is investigated in two cases. The first is where the medium is backed by a perfectly reflecting boundary. Here the intensity is shown to be a product of two factors; an exponential term that decays into the medium and the exponential of a ‘Brownian motion’ that describes the fluctuations of intensity with different realizations of the random medium. Because a Brownian motion has a normal distribution, the intensity has a log–normal distribution at any fixed point in the interior of the medium. For a typical realization of the random medium the exponential decay leads to most of the wave energy being near the front of a long medium. However, it is shown that the average intensity is independent of position in the medium. This is because of the long tail of the log–normal distribution and comes about because the average is heavily weighted by exceptional realizations of the medium. Thus the average value of the intensity, unlike the average of the logarithm of the intensity, is not representative of the intensity in a typical realization. The exponential decay of intensity is a result of the phenomenon of Anderson localization, which has received much attention in solid-state physics. The second case considered is where there is no barrier at the back of the medium. For large lengths of the random medium, it is shown that the transmitted intensity has an approximately log–normal distribution. The typical transmitted intensity is exponentially small as a result of localization, the decay rate with length being the same as the decay rate for the previous case. The average transmitted intensity is also exponentially small, but with a different decay rate because of weighting by exceptional realizations. The third problem discussed is that of the response of a random medium to time-harmonic forcing in the interior. The boundaries are taken to be perfectly reflecting and the response is found to be localized near the source for a typical realization. This result is related to the existence of localized normal modes in a long medium.