On the kinematics of broadband multipath scintillation and the approach to saturation

Utilizing a simple model in which the acoustic wave function is a sum of independent Gaussian wave packets, the relative intensity variance or scintillation index (SI) is analytically calculated. The model has an unspecified probability density function (PDF) for wave packet amplitudes and Gaussian PDFs for travel-time-induced and non-travel-time-induced phase shifts; amplitudes and both phase shifts are assumed to be mutually uncorrelated. It is shown that a proper treatment of the mean field is required to obtain the saturation value, SI = 1, in the limit of a large number of interfering wave packets. The analytic formulas for SI allow identification of important wave packet parameters in the approach to saturation. Criteria are identified for both broadband and narrow-band cases for which the approach to saturation is from above and below 1. It is demonstrated that the broadband approach to saturation is much slower than the narrow-band cases, since wave packets separated in time by more than an inverse bandwidth do not strongly contribute to interference. This effect is quantified by the time-bandwidth product. The model is also used to obtain an analytic expression for pulse time spread; it is shown that multipath conditions which favor a rapid approach to saturation do not favor large pulse spread.