On the applicability of high-frequency approximations to Lilleyʹs equation

Three forms of the high-frequency asymptotic Green function for Lilleyʹs equation are reviewed and compared to the exact solution over a wide range of Strouhal numbers. The asymmetric approximation, which applies to sources away from the jet axis, and the quasi-symmetric approximation, which uses a near-axis source assumption, are both obtained for parallel round jets from a formal Fourier-transform solution. The latter of the two is the basis of the so-called MGB computer code (High velocity jet noise source location and reduction, FAA-RD-76-96-II, 1978) and its derivatives. The ray-theory solution, which is the only high-frequency approximation that can be applied to more general mean flows, follows from a WKB ansatz and is shown to be closely related to the asymmetric approximation. The comparisons show that the best overall prediction of the exact Green function is given by the asymmetric approximation which remains accurate down to a Strouhal number of 1/2. The close relationship between the asymmetric and ray-theory approximations suggests that the high-frequency asymptotic Green function for more general mean flows would be similarly successful.