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

Abstract

Multiple wave scattering in strongly heterogeneous media is a very complicated phenomenon. Although a statistical approach may yield a considerable simplification of the mathematics, no guarantee exists that the theoretically predicted and the observed quantities coincide. The solution of this problem is to use self averaging quantities only. A multiple scattering theory that makes use of such self averaging quantities is the so called wave localization theory. This theory allows one to study both numerically and theoretically the influence of the presence of heterogeneities 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 phase velocities for several chaotic media described by different autocorrelation 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 results with estimates of the same quantities obtained from numerical simulations. The following can be concluded. (1) Theoretical predictions and numerical simulations agree in nearly the whole frequency domain for angles of incidence ≤30° and relative standard deviations of the fluctuations of the incompressibility ≤30 per cent. (2) An inspection of the inverse quality factor confirms that the apparent attenuation is strongest in the domain of Mie scattering except for fractal like media. In such media, no particular ratio of the wavelength to the typical scale length of heterogeneities is preferred since no such typical scale length exists. Hence, the inverse quality factor is constant over a large frequency band. (3) The group and phase velocities obtained agree with the effective medium theory and the Kramers–Krönig relations. That is, both converge to the effective medium velocity and the geometric velocity in the low and high frequency domains respectively. However, for intermediate frequencies, the exact behaviour strongly depends on the type of medium. Differences are related mainly to the number of extrema and Airy phases.