Alan R. Wenzel
NASA Langley Res. Ctr., MS 460, Hampton, VA 23681-0001
A theoretical analysis of the propagation of transient scalar waves in a one-dimensional random medium is presented. The index of refraction of the medium is assumed to deviate only slightly from unity, which allows the analysis to be carried out with the aid of a perturbation method. The specific approach adopted here combines a renormalization technique with a travel-time-corrected averaging procedure called asynchronous ensemble averaging. A general expression, valid for an arbitrary initial disturbance, is obtained for the variance of the wave. From that result, an expression for the variance is derived for the special case in which the initial disturbance has the form of a ramp function with arbitrary slope. These results show that the variance of the wave is directly proportional to the variance of the refractive index of the medium, but is only weakly dependent on the propagation path length. It is also found that, as the slope of the ramp function decreases, the wave variance decreases as well. The presentation concludes with some observations on the relevance of these results to the problem of sonic-boom propagation in the atmosphere. [Research supported by NASA.]