Acoustic wave propagation in one-dimensional random media: the wave localization approach

Multiple wave scattering in strongly heterogeneous media is a very complica ted phenomenon. Although a statistical approach may yield a considerable si mplification of the mathematics, no guarantee exists that the theoretically predicted and the observed quantities coincide. The solution of this probl em is to use self-averaging quantities only. A multiple scattering theory that makes use of such self-averaging quantiti es is the so-called wave localization theory. This theory allows one to stu dy both numerically and theoretically the influence of the presence of hete rogeneities on the frequency-dependent dispersion and apparent attenuation of a pulse traversing a random medium. I calculate the localization length (penetration depth), the inverse quality factor and both the group and phas e velocities for several chaotic media described by different autocorrelati on functions. Calculations are limited to 1-D acoustic media with constant density. However, media studied range from very smooth to fractal-like and incidence is not limited to be vertical. I then compare the theoretical res ults with estimates of the same quantities obtained from numerical simulati ons. The following can be concluded. (1) Theoretical predictions and numerical s imulations agree in nearly the whole frequency domain for angles of inciden ce less than or equal to 30 degrees and relative standard deviations of the fluctuations of the incompressibility less than or equal to 30 per cent. ( 2) An inspection of the inverse quality factor confirms that the apparent a ttenuation is strongest in the domain of Mie scattering except for fractal- like media. In such media, no particular ratio of the wavelength to the typ ical scale length of heterogeneities is preferred since no such typical sca le length exists. Hence, the inverse quality factor is constant over a larg e frequency band. (3)The group and phase velocities obtained agree with the effective medium theory and the Kramers-Kronig relations. That is, both co nverge to the effective medium velocity and the geometric velocity in the l ow- and high-frequency domains respectively. However, for intermediate freq uencies, the exact behaviour strongly depends on the type of medium. Differ ences are related mainly to the number of extrema and Airy phases.