Propagation of transients in a random medium

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Abstract
The propagation of transient scalar waves in a three−dimensional random medium is considered. The analysis is based on the smoothing method. An integro−differential equation for the coherent (or average) wave is derived and solved for the case of a statistically homogeneous and isotropic medium and a delta−function source. This yields the coherent Green’s function of the medium. It is found that the waveform of the coherent wave depends generally on the distance from the source measured in terms of a certain dimensionless parameter. Based on the magnitude of this parameter, three propagation zones, called the near zone, the far zone, and the intermediate zone, are defined. In the near zone the evolution of the waveform is determined primarily by attenuation of the high−frequency components of the wave, whereas in the far zone it is determined mainly by dispersion of the low−frequency components. The intermediate zone is a region of transition between the near and far zones. The results show that, in general, the randomness of the medium causes a gradual smoothing and broadening of the waveform, as well as a decrease in amplitude of the wave, with propagation distance. In addition, the propagation speed of the wave is reduced. It is also found that an oscillating tail appears on the waveform as the propagation distance increases.

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