Abstract
The wave propagation through a one dimensional inhomogeneous elastic media is studied from a mathematical view point. An ensemble of inhomogeneous elastic media is imagined and the wave propagation is investigated statistically. The wave field inherent in these media is separated into two distinct parts: the mean wave field and the fluctuation wave field. The former can be characterized by an ensemble average of the wave fields over all the possible random media. The randomness of the media is, herein, assumed to be stationary random functions of space with zero means and small compared to the extent. The statistical property of the ensemble is characterized by the cross correlation functions of the random inhomogeneities. The manner in which the mean wave field profile is modified by the scattering from the random inhomogeneities is discussed in detail. The equation governing the mean wave field up to the second order of the inhomogeneities is derived. The dispersion relation shows the gradual decay and the dispersive manner of the mean wave field in the media in which the inhomogeneities spread over a large region of space. The attenuation which characterize the exponential decay of the mean wave field is calculated for two examples and discussed.
