The phase‐random model of multipath acoustic propagation is used to derive the first‐order probability densities for the time rate‐of‐change of the short time average mean‐square pressure ?, and the time rate‐of‐change of the level in decibels of the short time average mean‐square pressure ?. It is shown that the probability densities for the signal amplitude, and amplitude rate are insensitive to frequency or phase modulation of the signal by the source, but that the density for the time rate‐of‐change of the multipath phase ?, is sensitive to such modulation. Because the finite bandwidths of acoustic signals can be modeled by uniform frequency modulation, the analysis presented applies to this problem as well. It is shown that bandwidth effects can be neglected only if B≪2ν, where B is the signal bandwidth, and ν2 is the single path mean‐square phase rate. This inequality provides a useful definition of what is meant by ’’narrow band’’ as it applies to phase‐random multipath propagation. It is also shown that the density for ? depends only on ν, while the density for ? depends on ν and parameters of the modulation. A potentially powerful technique is developed for determining source bandwidth and parameters of the source modulation, minus the effects of oceanic fluctuations, from the received multipath signal. The analytical results are compared with a computer simulation, and data from experiments in the ocean with extremely favorable results.