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General formulations for the fluctuations of a beam wave propagating through a homogeneous or locally homogeneous medium are given in terms of the spectral density of the index of refraction. The amplitude and phase correlation functions and the mean-square fluctuations are derived for a homogeneous medium showing the dependence on the radial distance in the transverse plane of the beam. The amplitude and phase structure functions are derived for a locally homogeneous medium. The correlation functions and the structure functions do not depend only on the difference coordinate; they are functions of the radial coordinates in the beam cross section. This particular inhomogeneity, however, is shown to be an analytic continuation of the homogeneous or locally homogeneous case. The meansquare amplitude fluctuation for the Kolmogorov's locally homogeneus medium is shown to behave as a plane wave for a short distance and then to become less than that of a spherical wave, and its spectrum is shown to behave as K−1 for large K in contrast with the plane and spherical waves. The spread of the beam radius is shown to be approximately the 8/3 powers of the distance L for small distance, and its, increase depends on the magnitude of the index of refraction fluctuation.