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Evolution of an oscillatory wide-band pulse in a sparse medium composed of randomly distributed, uncorrelated, discrete scatterers (such as atmospheric clouds, dust, or other aerosols) is studied. The frequency-dependent (dispersive) losses are evaluated by taking into account energy absorption in the medium constituents as well as scattering itself. A reduced, algebraic attenuation of the pulse energy is observed, provided the pulse contains a significant frequency content in the region of strongly varying medium dispersive properties. These frequencies can be provided by pulse carrier frequency selection, short rise and fall times of the pulse, or pulse chirping. It is shown that different types of algebraic attenuation, over the range of penetration depth corresponding to several orders of mean-free path, can be present depending on the inter-relations between characteristic frequencies of the pulse spectrum and the medium dispersive properties. A simple analytical model is constructed that captures relevant features of the propagating pulse energy decay, as well as ranges of penetration depths, and hence may serve as a useful tool in designing and analyzing various scenarios of wide-band pulse propagation in dispersive media in the context of, e.g., signal transmission, imaging, or target detection.