Correlation functions of a wave in a random distribution of stationary and moving scatterers

In this paper, integral and differential equations are derived for the correlation function of a wave in a random distribution of discrete scatterers. For stationary (nonmoving) scatterers, relatively simple integral and differential equations for the correlation functions are derived from Twersky's general formulations and their relationships with the radiative transfer theory are clarified. Small‐angle approximations of these differential equations are shown to be identical to the parabolic equations for the mutual coherence function in a turbulent medium. For moving scatterers, Twersky's formulation is modified to take into account the particle motion and the variation of the field in time. General integral and differential equations for spatial as well as temporal correlation function are obtained which include the effects of the constant velocity as well as the fluctuating velocity. Differential equations governing the temporal frequency spectra of the field as well as the specific intensity are derived showing the Doppler shift and the spectrum broadening.