Humming hummingbirds, insect flight tones, and a model of animal flight sound

Why do hummingbirds hum and insects whine when their wings flap in flight? Gutin proposed that a spinning propeller produces tonal sound because the location of the center of aerodynamic pressure on each blade oscillates relative to an external receiver. Animal wings also move, and in addition, aerodynamic force produced by animal wings fluctuates in magnitude and direction over the course of the wingbeat. Here we model animal wing tone as the equal, opposite reaction to aerodynamic forces on the wing, using Lowson's equation for the sound field produced by a moving point force. Two assumptions of Lowson's equation are met: animal flight is low (< 0.3) Mach and animals from albatrosses to mosquitoes are acoustically compact, meaning they have a small spatial extent relative to the wavelength of their wingbeat frequency. This model predicts the acoustic waveform of a hovering Costa's Hummingbird (Calypte costae), which varies in the X, Y, and Z directions around the animal. We modeled the wing forces of a hovering animal as a sinusoid with an amplitude equal to body weight. This model predicts wing sound pressure levels below a hovering hummingbird and mosquito to within 2 dB; and that far-field mosquito wing tone attenuates to 20 dB within about 0.2 m of the animal, while hummingbird humming attenuates to 20 dB at about 10 m. Wing tones play a role in communication of certain insects, such as mosquitoes, and influences predator-prey interactions, because it potentially reveals the predator's presence to its intended prey.