The propagation of transient scalar waves in a three‐dimensional random medium is considered. The analysis is based on the smoothing method, which is equivalent to a second‐order scattering theory. An integro‐differential equation for the coherent (or average) wave is solved for the case of a statistically homogeneous and isotropic medium and a delta‐function source. This yields the coherent Green's function of the medium. It is found that, in general, the randomness of the medium causes a gradual smoothing and broadening of the waveform with propagation distance, as well as a reduction in propagation speed of the wave. Depending on the distance from the source measured in terms of a certain dimensionless parameter, three propagation zones can be identified. In the near zone the waveform is determined primarily by attenuation of the high‐frequency components of the wave, whereas in the far zone it is determined mainly by dispersion of the low‐frequency components. The intermediate zone is a region of transition between the near and far zones. The theory yields order‐of‐magnitude estimates of sonic boom rise times which show good agreement with measured values.