This paper is concerned with the small-amplitude unsteady motion of an inviscid non-heat-conducting compressible fluid on a transversely sheared mean flow. It extends previous analyses (Goldstein, J. Fluid Mech., vol. 84, 1978b, pp. 305–329; Goldstein, J. Fluid Mech., vol. 91, 1979a, pp. 601–632), which show that the hydrodynamic component of the motion is determined by two arbitrary convected quantities in the absence of solid surfaces and hydrodynamic instabilities. These results can be used to specify appropriate upstream boundary conditions for unsteady surface interaction problems on transversely sheared mean flows in the same way that the vortical component of the Kovasznay (J. Aero. Sci., vol. 20, 1953, pp. 657–674) decomposition is used to specify these conditions for surface interaction problems on uniform mean flows. But unlike Kovasznay’s result, the arbitrary convected quantities no longer bear a simple relation to the physical variables. A major purpose of this paper is to complete the formalism developed in Goldstein’s earlier two papers by obtaining the necessary relations between these quantities and the measurable flow variables. The results are important because they enable the complete extension of non-homogeneous rapid distortion theory to transversely sheared mean flows. Another purpose of the paper is to derive a generalization of the famous Ffowcs Williams and Hall (J. Fluid Mech., vol. 40, 1970, pp. 657–670) formula for the sound produced by the interaction of turbulence with an edge, which is frequently used as a starting point for predicting sound generation by turbulence–solid surface interactions. We illustrate the utility of this result by using it to calculate the sound radiation produced by the interaction of a two-dimensional jet with the downstream edge of a flat plate.