A theoretical analysis of the wavefield radiated by a point source in a weakly inhomogeneous one‐dimensional random medium is presented. The analysis is based on a perturbation technique, and includes all single‐ and double‐scattering terms, as well as some scattering terms of higher order. An expression for the mean intensity (i.e., the mean square of the modulus of the complex wave amplitude) of the wave as a function of source‐receiver distance, or propagation distance, has been obtained. This expression shows that, inside a certain region of space containing the source, the mean intensity of the wave is increased by the randomness of the medium, whereas outside of this region it is decreased. The result of this redistribution of mean intensity in space is a more rapid rate of decrease of the mean intensity (or the log of the mean intensity) with propagation distance than would be the case in the absence of randomness, which can be interpreted as an excess attenuation of the wave as a result of the randomness of the medium. The relation of the present theory to previous theories of attenuation of waves by randomness, or turbulence, is briefly discussed. [Work sponsored by NORDA.]