This paper describes a numerical methodology for calculating tonal noise propagation and radiation through turbomachinery exhaust ducts, including non-uniform background jet flows. The numerical method is based on solution of the linearized Euler equations directly in the frequency domain, employing a direct, sparse matrix solver in parallel. Acoustic sources are introduced into the computational domain via the perfectly matched layer equations. Various test cases including propagation through infinite ducts, and propagation and radiation through semi-infinite ducts with and without liners are solved, and results are compared with analytical solutions. It is demonstrated that solutions even with thin shear layers separating the exterior and jet streams at reasonably high Mach numbers can be obtained quite successfully. The method is also validated by simulating the radiated sound waves from two actual engine exhaust cowl geometries, and the results are compared with experimental data. (C) 2009 Elsevier Ltd. All rights reserved.