Fractional precursors in random media

Abstract

When a broadband pulse penetrates into a dissipative and dispersive medium, phase dispersion and frequency-dependent attenuation alter the pulse in a way that results in the appearance of a precursor field with an algebraic decay. We derive here the existence of precursors in nondispersive, non-dissipative, but randomly heterogeneous and multiscale media. The shape of the precursor and its fractional power law decay with propagation distance depend on the random medium class. Three principal scattering precursor classes can be identified : (i) In exponentially decorrelating random media, and more generally in mixing random media, the precursor has a Gaussian shape and a peak amplitude that decays as the square root of the inverse of the propagation distance. (ii) In short-range correlation media, with rough multiscale medium fluctuations, the precursor has a skewed shape with a tail that exhibits an anomalous power law decay in time and a peak amplitude that exhibits an anomalous power law decay with propagation distance, both of which depend on the Hurst exponent that characterizes the roughness of the medium. (iii) In long-range correlation media with long-range memory, the situation mimics that of class (ii), but with modified power laws.

Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography

“…( 37) and the random effective propagation model Eq. ( 16) in Garnier and Sølna (2010). Näsholm and Holm (2011) considered the integral generalization…”

Simple asymptotic forms for Sommerfeld and Brillouin precursors

“…(24) and (25) are not specific to the Lorentz medium but have some generality [37]. They hold for the Debye medium [38], for some random media [39], and, more generally, whenever the transfer function of the medium can be expanded in cumulants and the propagation distance is such that |κ| 1. Stoudt et al [38] showed in particular that the results of their experiments on water (Debye medium) at decimetric wavelengths can be numerically reproduced by neglecting the group-delay dispersion, that is the approximation made to obtain the analytical result of the Eq.…”