![]() Tones with low and zero azimuthal mode order are treated as special cases and the asymptotics show that, as the mode order reduces, the radiated sound becomes concentrated around the flight axis where even higher-order solutions are possible, including rings and annuli of stationary points around the propeller annulus. The paper also shows how the interior points are completely consistent with the sub- or super-critical gust response of a swept blade. The asymptotics show that sweep is not necessarily beneficial and can cause the blade design to become critical for particular tones and directions in terms of a continuum of interior points distributed along a line on the propeller source annulus producing a higher-order result and thus an enhanced radiated sound field. The asymptotic approach is powerful producing results in the form of one-line algebraic formulae that contain no integrals or special functions yet remain accurate. Assuming that the number of blades on both propellers is large, we evaluate the integral asymptotically in terms of its leading-order contributions from interior stationary or boundary critical points which represent the specific locations on the propeller annulus that dominate the sound radiation. ![]() We re-cast the usual far-field radiation formulae as a double integral over a nominal propeller source annulus. An analytical model is presented for the wake interaction tones produced by a contra-rotating propeller. ![]()
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