In a previous paper the author treated the two-dimensional problem of a plane sound wave impinging on a cascade of flat-plate airfoils of infinite span in a stream of finite subsonic Mach number, and with a chord and spacing small compared to the acoustic wavelength.
In the present study this problem is extended to the case of sound propagating upstream through two blade rows (e.g., a rotor-stator combination). The viscous wakes of the upstream blade row are neglected, but the vorticity shed from the upstream blade row due to the influence of the incident acoustic wave is accounted for. This vorticity interacts with the downstream row as does the acoustic wave reflected from the upstream row. The airfoil spacing and chord of both blade rows are assumed to be small compared to the acoustic wavelength of the incident wave. The distance between the two blade rows is arbitrary with respect to the wavelength, but large compared to the spacing between the airfoils of either row.
The assumption of long wavelength makes possible the derivation of a closed-form solution. Results for transmitted and reflected pressure are given for various blade row geometries. The spacing between the blade rows is found to be an important parameter in the solution, but resonances depending on this spacing were not found.