The propagation of random sound in a cylindrical duct has been treated theoretically. The conditions under which the simple wave equation applies to the transmission of sound generated by industrial air‐moving devices is made explicit. The acoustic forcing function and termination conditions are discussed extensively as well as the structure of the sound field in the duct. Equations for the power spectral density and pressure cross‐spectral density are derived in order to define the problems associated with deducing sound power, given sound pressure at two points. The results suggest that an anechoic termination is almost mandatory and that the modal spectra must be known, if accurate measurements are to be made. A number of simplifications to the theory were tested experimentally and found to apply. Modal spectra were measured and were used to compute sound power and predict pressure measurements. The predicted and observed were found to be within 1 dB, suggesting that a practical means for measuring ducted‐fan sound power can be found.