The problem of the propogation of harmonic signals in a slightly rough duct is investigated. The analysis made tractable by the assumption that the heights of the roughness peaks are small compared to the average separation of the duct walls. Both acoustic and electromagnetic waves are treated. In the case of a duct with harmonic roughness, it is found that waves are not scattered upstream if the frequency of the transmitted signal exceeds certain critical values related to the natural modes of the wave guide and the wavelengths of the wall roughness. It is also found that the scattered signals are amplified whenever the frequency of the transmitted signal is equal to any of a series of critical frequencies determined by the natural modes of the guide and the spectrum of the wall roughness. Statistically rough walls are also treated, and specific results worked out for roughness functions having Dirac delta function correlation exponential correlation.