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The so-called parabolic equation (PE) has proved its capability to deal with the long range sound propagation as an alternative to the ray model. It was shown that the High-Order Parabolic Equation (HOPE), based on a Pade expansion, significantly increases the aperture angle of propagation, compared to the standard PE and the wideangle PE. As a result, for the in-duct propagation it allows us an accurate prediction close to the cut-off frequency. This paper concerns the propagation using the HOPE in heterogeneous flows, including a boundary layer above a hard wall and in shear layers. The thickness of the boundary layer is some dozens of centimeters while, outside of it, the Mach number can reach 0.5. The flow effects are investigated showing the refraction effects at a propagation distance of 30 meters, up to a few kilohertz. Significant discontinuities in the directivity patterns occur in the shear layer. Comparisons with the Euler solution are considered, including configurations beyond the theoretical limits of the HOPE.

Capabilities of the high-order parabolic equation to predict sound propagation in boundary and shear layers.