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

The whine of a mosquito right by one's ear may be among the most universal of human experiences. Flight tones of certain midge and mosquito species serve a communication role in courtship (Cator et al., 2009; de Silva et al., 2015). Other small animals make wing tones in flight, such as the buzzing of bees and flies, or the humming of hummingbirds and hawkmoths as they fly from flower to flower, for which a clear function has not been identified. Flight tones seem to be a ubiquitous feature of flapping wings. Most prior work on insect and hummingbird flight tones has centered on the observation that the dominant (highest amplitude) frequency of the flight tone is either the wingbeat frequency (Arthur et al., 2014; Sotavalta, 1952; Sueur et al., 2005; Webb et al., 1976; Williams and Galambos, 1950) or, from certain angles, double the wingbeat frequency (Bae and Moon, 2008; Dudley, 2000). Here, we explored the question: why does a flapping wing produce tonal sound that varies with orientation around the animal?

Acoustic theory developed for propellers suggests multiple possible mechanisms (Inada et al., 2009; Seo et al., 2019). A spinning rotor produces three types of sources of sound: monopole-like ‘unsteady source of mass’ (Bodony et al., 2016), where sound is produced by a volumetric process; dipole-like ‘load’ or Gutin sound (Fig. 1A), where sound is produced by a force; and quadrupole-like ‘broadband’ sound (Blake, 2017a; Howe, 2008; Magliozzi et al., 1981; Schmitz, 1981).

This third broadband noise term, on a wing, is primarily caused by ingestion, formation and scattering of turbulence and vorticity (at the leading edge, surface and trailing edge, respectively) (Blake, 2017b). Such sounds are widespread in animal flight, even in the relatively quiet flight of owls (Sarradj et al., 2011). But as this sound source type is broadband (atonal), it is not consistent with wing tone that is the subject of this paper, and cannot be its source.

Regarding the first monopole-like term, in a propeller spinning at a constant angular velocity, one type of volumetric process that generates sound is thickness noise, caused by the sudden (unsteady) volumetric displacement of air by the propeller as it passes through a control volume, where the air is displaced in proportion to the propeller's thickness. The effect of thickness noise is reduced at lower Mach numbers (Blake, 2017a; Glegg and Devenport, 2017). As animal flight is low Mach (see below), whether thickness noise could be a significant source of animal wing sounds is unclear. Other monopole-like sources encompassed by an ‘unsteady source of mass’ that are not present in a propeller spinning at constant velocity might nevertheless be present in animal flight (Bodony et al., 2016). For instance, in ordinary hovering, a bulk of air Q(t) is moved from above the animal to below the animal with each flap of the wings (Fig. 1A). If this induced bulk flow were continuous (as it is across a propeller spinning at a constant rate), it would not produce sound (Lighthill, 1962). But because animal wings reciprocate, rather than rotate at a constant rate like a propeller, the movement of the bulk of air is unsteady rather than continuous, and Q fluctuates in pulses associated with each wingbeat (Fig. 1A). As the time derivative (dQ/dt) is not zero, this mechanism hypothetically might contribute to the sound produced by animal wings (Bodony et al., 2016).

Of the three source types, propeller theory suggests a likely source of wing tone is the second type of sound, the reaction to fluctuating aerodynamic force (Glegg and Devenport, 2017). Gutin (1948) was the first to identify lift and drag as a major source of tonal sound on rotating propellers. The blades of a propeller, if spinning at a constant rate in uniform air, each develop force, F(i,t), that is invariant in a local frame of reference. The equal, opposite reaction to this aerodynamic force imparted on the wing is aerodynamic pressure, P(i,t), exerted on the surrounding air (Fig. 1B), a manifestation of Newton's 3rd law (Blake, 2017a). As the force is invariant, so is the pressure. An observer spinning in the frame of reference of the propeller would not experience any fluctuation in pressure, and hence hears nothing. But Gutin (1948) reasoned that, as the location of the center of pressure on each blade oscillates relative to an external receiver, this external receiver experiences fluctuating pressure associated with the source's oscillatory motion. Because thrust is perpendicular to the plane of the spinning propeller, while drag is parallel to this plane, a receiver in front or behind the plane of the propeller hears sound generated in reaction to the rotating sources of thrust (except exactly on the central axis of the propeller, where the distance between the center of pressure and the receiver does not fluctuate, and thus there is no sound). A receiver to the side (in plane with the propeller) instead hears sound that is the aerodynamic reaction to drag (Blake, 2017a). This Gutin sound has a property of wing tone: it is tonal and varies with orientation around a propeller.

The wings of flying animals are similar to propellers: the wing moves (black solid vectors in Fig. 1C). Additionally, unlike the idealized propeller just described, the aerodynamic force vector on flapping wings (Fig. 1B, blue vectors) itself also varies substantially in magnitude and direction over the course of the wingbeat (Sane and Dickinson, 2001). The time derivative of force, δF(i)/δt and the corresponding pressure, δP(i)/δt, are not zero. Thus, a flapping animal wing may produce sound during flight, caused by fluctuations in magnitude, direction and location of F(i,t) on the wing, which generate fluctuations in pressure that a stationary receiver perceives as sound (Fig. 1C).

Here, we modeled this Gutin sound produced by the wings of a hovering hummingbird, using Lowson’s (1965) equation for the sound field produced by a moving point force. Recent models of insect wing sound (Inada et al., 2009; Nedunchezian et al., 2019; Seo et al., 2019) have instead used the Ffowcs Williams–Hawkings (FW–H) equation (Ffowcs Williams and Hawkings, 1969). The FW–H equation was a refinement of Lowson's equation, as Lowson's equation becomes singular as speed approaches Mach (M)=1, while the FW–H equation does not suffer from this limitation. The FW–H equation provides the sound field produced by a moving surface pressure distribution, rather than Lowson's moving point force. While the FW–H equation is accurate as speed approaches M=1, using it requires the time history of the pressure distribution on a source, for instance, as computed with a computational fluid dynamics (CFD) model of the aerodynamic forces on an animal's wing (Seo et al., 2019). Lowson's equation instead requires an estimate of a point force (such as on the center of pressure of the wing).

Lowson's assumption of low Mach was not safe for propellers, but is safe for animal flight, where low Mach means M<0.3 (Glegg and Devenport, 2017). The Mach number is non-dimensional speed relative to the speed of sound (c), which for room-temperature air is roughly 340 m s⁻¹. The fastest flying extant animal is a falcon diving at 70 m s⁻¹ (Tucker, 1998), which is a speed of Mach ∼0.2, while the wingtip of a hovering hummingbird reaches roughly ∼10 m s⁻¹, or Mach 0.03 (Clark and Mistick, 2018). Insect wingtips flap at even lower Mach numbers. Only one (extinct) biological structure has been suggested to have been supersonic: the whipped tip of a sauropod tail (Myhrvold and Currie, 1997). Aerodynamic models that are only valid at low Mach numbers are likely valid for nearly all biological applications.

Lowson's other important assumption was that his point force was acoustically compact, with a Helmholtz number of ≪1. This Helmholtz number, also called the ka product, is a dimensionless number that describes sound source size relative to the wavelength of sound it produces (Larsen and Wahlberg, 2017). The parameter a is a characteristic source radius (treating the source as a sphere), such as animal wing length; and k is the wavenumber, k=2πn/λ, where λ is the wavelength associated with the wingbeat frequency (f) (λ=c/f) (Larsen and Wahlberg, 2017), and n is an integer harmonic of the wingbeat (n=1 is the fundamental, n=2 is the 2nd harmonic, etc.).

Here, we used wing hum of a hovering hummingbird as a test case for this model, as hummingbirds have wing hum that is salient to humans, and have been the subject of a CFD model of wing force in the three cardinal directions around the hovering animal (Song et al., 2014). Then, we generalized the model for the sound field directly below a hovering animal, by assuming that hovering animals produce aerodynamic force sinusoidally with an amplitude equal to body weight. We assessed this model against literature values for bird and insect wing sounds. Finally, we discuss the biological implications (communication and predator–prey interactions) of this sound field generated by a flapping wing.

The left-hand term containing δQ/δt of Eqn 1 is the unsteady source of mass term described previously. The central and right terms are the Gutin sound. The central term containing F_i is the nearfield generated by the aerodynamic force (eqn 18 in Lowson, 1965), which scales as r⁻², as (x_i−y_i) is proportional to r. The right-hand term containing δF_i/δt is farfield pressure, which scales as r⁻¹.

To test this model's assumption that ka<<1, wing length, f, and body mass were obtained from the literature for a range of birds and insects (Greenewalt, 1962; Pennycuick, 1990).

We tested our model by comparing the wing hum of a hovering hummingbird with the predicted wing hum from a CFD model. Song et al. (2014) used a CFD model to estimate the non-dimensional wing forces F(i,t) of a hovering male Ruby-throated hummingbird (Archilochus colubris; assuming body mass: 2.7 g, wingbeat frequency: 57 Hz) in the three cardinal directions around the bird (i=x, y, z). We re-scaled their data (in amplitude) to produce weight support in z for a female Costa's hummingbird (Calypte costae: body mass: 3.2 g, wingbeat frequency: 37 Hz). We then inserted the resulting estimate of F(i,t) from Song et al.’s (2014) CFD model into Eqn 2 to generate a predicted nearfield P(i,t) for a hovering female Costa's hummingbird.

A female Costa's hummingbird, Calypte costae (Bourcier 1839), was recorded as she hovered, stationary, at a feeder inside a cage. The bird was released after recording. Recordings were conducted under UCR IACUC protocol 20160039 and USFWS and CDFW collecting permits. These recordings were taken in the middle of a small room (approximately 5×5×3 m) in a building lacking HVAC equipment at the Boyd Deep Canyon Natural History Reserve (doi:10.21973/N3V66D). Recordings were made on a windless day, after switching off sound sources (such as computers), minimizing environmental background sounds. Thus, although this space was not anechoic, it provided low background sound levels (<20 dB re. 20 μPa), in which the bird's wing hum was the dominant sound within the room. The recordings were taken with two ½ inch free-field microphones (Brüel & Kjaer 4190; Nærum, Denmark) with a noise floor of 14.6 dB and sensitivity of 50 mV Pa⁻¹. The microphone below the bird was outfitted with a turbulence-reducing nose cone (Brüel & Kjaer UA 0386), which had no effect on its low frequency response, and no further acoustic shielding. Recordings from below the bird in which the bird's wake impinged directly on the microphone with audible microphone-induced noise were not used. The recordings were made on a 24-bit recorder sampling at 96 kHz (Sound Devices 702; Reedsburg, WI, USA). The recorder's lowest frequency was 10 Hz, whereas the microphone had a flat frequency–response curve that extended below this value. The two microphones were placed in different configurations, 0.15 to 0.21 m from the bird (i.e. in the nearfield), to record from the three cardinal directions (during different bouts of hovering). The floor and walls were >1 m from the subject, such that reflected farfield sound traveled at least 2 m before returning to the microphone, and thus was ≥20 dB below directly transmitted sound (assuming it diminished as r⁻¹).

Sound pressure level (SPL) was calculated from power spectra of sounds in Raven 1.4 (Charif et al., 2008) with a FFT window of 65,536 samples. SPL was calculated relative to calibration recordings of known SPL produced with a B&K 4231 sound level calibrator (re. 20 μPa) (Clark et al., 2013).

During hovering, an animal produces sufficient aerodynamic force oriented on its z-axis to exactly offset its body weight (mass×gravitational acceleration, mg, ∼10 m s⁻²). Hence, at distance r below the hovering animal, there is an average increase in pressure above ambient P₀, where [] is proportional to body weight (mg) and inversely proportional to distance (Eqn 3) or distance squared (Eqn 2). As microphones are high-pass instruments, if the animal hovers for a ‘long’ time (such as >1 s), a microphone with a minimum frequency of 10 Hz will quickly lose the initial transient change in mean pressure from P₀ to associated with the onset of hovering (Fig. 4A). That is, a microphone cannot measure directly. can instead be estimated from the time course of force production by the wings P_hov(z), by assuming that during hovering the fluctuations in P_hov(z) around are similar in magnitude to ⁠.

From eagles to mosquitoes, the Helmholtz number of birds and insects is invariant with body size (Fig. 2). Within the first six harmonics (i.e. n=1…6) of their wingbeat, birds and insects are acoustically compact; the Helmholtz number of the fundamental frequency (1f) varied between 0.02 and 0.07 among most animals sampled (Fig. 2B) (the greatest outliers are butterflies, with their low wingbeat frequencies, where ka≈0.01). These data show that birds and insects have a maximum spatial extent that is much smaller than the wavelength of sound produced by their flapping wings. Because animals are compact sound sources, interference between the wings is low, while the ‘reactive nearfield’ (nearfield effects caused by high particle velocity magnitudes) will be substantial (Larsen and Wahlberg, 2017).

The functional form of the predicted P(i,t) from Song et al.’s (2014) CFD model is a qualitative match to our measured P(i,t) of a female Costa's hummingbird in all three dimensions (Fig. 3). The predicted nearfield wing sound derived from Song et al.’s (2014) model, scaled to a unit distance of r=1 m, is shown in Fig. 3A, with downstroke depicted with gray shading (note: recordings were taken at distances of 0.15–0.21 m; had we recorded at 1 m distance, farfield sound would have not been negligible). Three representative acoustic traces (from two different bouts) of the hovering Costa's hummingbird are shown in Fig. 3B. Recordings of hummingbird wing hum from the three cardinal directions are included in Audio 1–3. Wing sounds were at the highest amplitude in the 1st harmonic in the x direction, the 2nd harmonic was higher in the y direction, and in z, the amplitude of the first two harmonics was similar, 62.6 and 63.1 dB (Fig. 3C). Cumulatively the first six harmonics in the z direction sum to 70.5 dB (r=0.21 m); harmonics above the 6th were sufficiently low amplitude (in all three dimensions) that we ignored them. Two wingbeats (onset of downstroke aligned by eye) are shown in Fig. 3D, while Fig. 3E shows a comparison of model and data. Quantitatively, the amplitude of the sound recording in the i=x and z directions is 1.75 times (2 dB) the amplitude of predicted nearfield values from our model.

Eqn 4 predicts the Gutin sound below the animal, from its body mass and wingbeat frequency. How far does this model predict wing sound will propagate? A mosquito flapping its wings at 1 kHz would be faintly audible at 20 dB (re. 20 μPa) against the background sound of a quiet room. Therefore, we defined the 20 dB distance as the distance r at which the farfield P_hov reaches 20 dB (Fig. 5A) (at r<<λ/2π, the nearfield term is negligible). Among the birds and insects sampled, the 20 dB distance varied primarily as a function of body weight, because weight determines the total aerodynamic force in z the animal needs to produce with its wings to remain aloft. Albatrosses or eagles have wingbeat frequencies as low as 2 Hz, so would have 20 dB distances of up to 1 km, while among the smaller animals capable of hovering, hummingbird wing hum (f of 25–100 Hz) is predicted to reach 20 dB approximately 10 m from the animal. Bees (e.g. Bombus spp., f: 200 Hz) have wing sounds that reach 20 dB at distances of 1–4 m, while mosquito wing sounds (f: 400–1000 Hz) are predicted to reach 20 dB at roughly 0.1–0.3 m (Fig. 5).

The nearfield sound amplitude is proportional to mg irrespective of f, thus the curves of predicted sound levels for different animals in Fig. 5B are strictly in order of body mass. Farfield sound is proportional to mfg (Eqn 4); therefore, although body mass is a strong determinant of the functional form shown in Fig. 5C, the curves are not strictly in order of body mass. Species with f that is low for their body mass (e.g. butterfly, Pieris napi) have proportionally lower amplitude of farfield sound and, conversely, species with high f for their body size (e.g. mosquito, Aedes aegypti) have elevated farfield sound (Fig. 5C). This farfield dependence on f also means that harmonics above the fundamental contribute disproportionately to farfield sound (Fig. 6B).

The empirical measurements of wing SPL we found in the literature are presented in Fig. 6 and Table S1. Among these, wing sounds have been recorded on insects that are tethered or perched (e.g. honeybee waggle dance wing sound) and therefore do not support their weight while flapping. Arthur et al. (2014) recorded wing tone from a tethered mosquito at distances spanning the nearfield–farfield transition. Their data in the z direction are a good quantitative fit to our model (Fig. 6A), as are the shapes of their waveforms in x and y (fig. 5A of Arthur et al., 2014).

Boonman et al. (2018) recorded wing sound from underneath a barn owl taking off. Their sound amplitudes are substantially below the predicted values in z (Fig. 6C). LePiane and Clark (2020) recorded from underneath barn owls as they flew over a microphone, with values closer to those predicted by the hovering model, with an exponent of −1.1. Boonman et al. (2020) recorded wing sounds of four bird and three bat species in forward flight, from an oblique angle which changed with distance. Their data and our prediction for sound in z are shown in (Fig. 6D); their empirical exponent varied between −1.0 and −2.3 among the species measured.

We have developed a simple model for the sound field below an animal made by the fluctuating aerodynamic forces of flapping wings of a hovering animal. Nearfield recordings of the humming of hummingbirds and mosquitoes are consistent with this relatively simple model, which ignored the relative motion of the wings (we set M=0), assumed force was produced sinusoidal, ignored contributions of thickness noise or unsteady induced flow (the δQ/δt term in Eqn 1) (Figs 3 and 6), ignored retarded time (which is the propagation delay between source and receiver), and ignored the presence of more than one wing.

The most general prediction this model makes seems to answer the question posed in the Introduction: one source of wing tone is the time course of aerodynamic force production of the wings. As the wings reciprocate back and forth and are rotated at the end of the stroke, they produce an aerodynamic force vector that fluctuates in magnitude and direction. The aerodynamic pressure response therefore also fluctuates in each of the cardinal directions around the animal (x, y, z) in equal and opposite reaction to the wing aerodynamic forces. Aerodynamic effects that predominate on insect wings, such as the lift produced by wing rotation and vortex shedding, are intrinsically included in this model, as these mechanisms produce aerodynamic force and are not intrinsically different from other sources of lift and drag (Walker, 2002). For instance, periodic vortex shedding produces tonal sound, but not as a separate acoustic mechanism, because the way vortex shedding produces sound is by producing periodic cycles of lift and drag on the solid object (Blake, 2017a; Blake, 2017b, chapter 4). This mechanism provides a general explanation for why the flight tone in certain directions is, in many species, loudest (dominant) at the fundamental frequency of the wingbeat, while in other directions the flight tone is dominant at the 2nd harmonic of the fundamental frequency of the wingbeat, as previously noted in flies (Sueur et al., 2005), mosquitoes (Arthur et al., 2014), other insects (Sotavalta, 1952), and shown here for hummingbirds (Fig. 3). Which anatomical directions are fundamental-dominant versus 2nd harmonic-dominant depends on the time course of force production of the species in question.

An interesting theoretical case are four-winged insects such as dragonflies, in which ipsilateral (fore and hind) wings reciprocate with some independence, and flap at the wingbeat frequency but out of phase with each other. Independently reciprocating wings are similar to propellers that have multiple independent blades. The Gutin sound produced by a propeller is a collective product of all of the blades; hence, the frequency produced is the blade passage frequency, which is the rotation rate multiplied by the number of blades (Blake, 2017a; Gutin, 1948). In a dragonfly, assuming fore and hind wings produce similar amplitudes of lift, their wing tone is predicted to be at the even harmonics (2, 4, 6, etc.) of the wingbeat frequency, because their ‘wing passage frequency’ is double their wingbeat frequency.

Arthur et al. (2014) present detailed data on the wing sounds of tethered mosquitoes to which our model is a good fit (Fig. 6A). Many of the precise empirical details they report for mosquito wing tone are likely to be general to animal wing tones. They report that the phase relationships between the harmonics of wing sound are precise and fairly invariant (their Fig. 5). The flight stroke of flying animals has precise kinematics (such as timing of wing rotation) tuned to maximize useful aerodynamic forces (usually lift) and minimize not-useful forces (usually drag) produced by the wings. The waveforms of most hovering animals seem likely to resemble those shown in Fig. 3 (except for the dragonfly case described above). Deviating substantially from these phase relationships would produce an animal with mistimed wing kinematics that would tend to fail to produce sufficient aerodynamic force to remain aloft, or would produce high drag. This is not to say the phase relationships would be completely invariant; flying animals maneuver (turn, accelerate) by producing unbalanced forces with their contralateral (left, right) wings. This is achieved by subtly modifying the kinematics (varying amplitude and timing) of one wing relative to the other (Altshuler et al., 2012), which has the effect of causing phase between the harmonics to vary subtly. Moreover, as tethered animals do not support their body weight and may attempt to maneuver in ways that likely vary in time and between animals, completely invariant phase relationships in recordings of a tethered animal are not expected.

Arthur et al. (2014) suggested that their mosquito wing tone attenuated slightly faster than the square of distance (exponent of −2.3 to −2.5) over the distances at which they measured sound (their data are re-plotted in Fig. 6A). Our model suggests the rate of attenuation of the nearfield Gutin sound in the z direction is with the square of distance (αr⁻²), and in the farfield (r>0.07 m), attenuates as r⁻¹. The slight discrepancy in exponent between Arthur et al.’s (2014) data and our model could be caused by an additional source of pressure close to the animal that decays with a higher exponent. Alternately, it could be caused by slight measurement error. Overall, their data are a good fit to the predicted exponent of −2 (Fig. 6A). Our model provides a fit to Arthur et al.’s (2014) data that is similar to the FW–H model of Seo et al. (2019) of Arthur et al.’s (2014) data (dashed lines in Fig. 6A). The mismatch between the model and empirical data for birds and bats (Fig. 6C,D) has multiple simple explanations. Boonman et al.’s (2018, 2020) data are lower than the predicted values, but they recorded birds and bats with microphones with frequency–response curves that were flat down to 50 Hz (Boonman et al., 2018) and 40 Hz (Boonman et al., 2020), respectively, above the wingbeat frequencies of the species they measured. Moreover, Boonman et al. (2020) recorded from an oblique angle, while Fig. 6D shows the model's prediction in z. Although the Gutin sound in x and y of mosquitoes or hummingbirds is of the same order of magnitude as the Gutin sound in z (Fig. 3; Arthur et al., 2014), this did not seem to be the case for barn owls in forward flight (LePiane and Clark, 2020). If the Gutin sound of birds or bats is more directional than that of insects, it could provide another simple reason why Boonman et al.’s (2020) measured sound levels are well below the model predicted values in z. The model is for hovering, but both LePiane and Clark (2020) and Boonman et al. (2020) measured animals as they flew past a single microphone. In both these studies, orientation co-varied with distance, which might explain why the exponent among the species shown in Fig. 6C,D varied from −1.0 to −2.2.

The model presented here is a minimum model: animals that produce fluctuating aerodynamic forces with their wings to remain aloft must produce the fluctuating pressure modeled here (subject to the assumption about lift shown in Fig. 4C). Flying animals may produce additional sounds, via a variety of mechanisms. For instance, our model seems to adequately describe the hum of hummingbirds, but it appears to not entirely explain another familiar insect wing sound: the buzzing flight sound of bees and large flies. To human ears, Gutin sound is like a hum, because wing motions can be adequately described by roughly the first six harmonics of motion (Fig. 3C; Bae and Moon, 2008). The buzzing characterizing the wing sound of bees and flies arises from the presence of much higher harmonics (far above 6f). It is these higher harmonics that makes them sound buzzy, rather than a hum, to human ears. These higher harmonics are not predicted by our model, implying they are not Gutin sound. They may not be produced the wings: some bees engage in ‘buzz pollination’ in which they audibly make buzzing sounds that resemble their flight sound when their wings are folded (Burkart et al., 2011), suggesting that one major source of the buzzing of bees may be the thorax, rather than the wings.

The sound modeled here is the equal, opposite aerodynamic reaction to the production of aerodynamic force, the aerodynamic equivalent to ground reaction forces produced during terrestrial locomotion. Use of force plates to measure the ground reaction forces of animal appendages is widespread in studies of terrestrial locomotion (Full and Tu, 1991). A similar technique for flyers would open up a range of new topics of study on birds, bats and, especially, insects. Lentink and colleagues have developed an ‘aerodynamic force plate’ (AFP) that measures the instantaneous total aerodynamic force when an animal flies above it (Chin and Lentink, 2019; Hightower et al., 2017; Lentink et al., 2015), by physically integrating pressure across the entire surface area above and below the animal. Limits to Lentink et al.’s (2015) method include the need to record on both sides of the animal within a confined space, the inability to resolve force produced by one wing relative to the other and, especially, the limits on the devices' ability to record the more subtle pressures produced by small animals (i.e. insects).

Whether the dynamic range of microphones might overcome this last limitation and permit direct estimates of flight force from wing sound in ways the AFP cannot is not immediately clear. Several assumptions of our model require further refinement. In the comparison between model and prediction in Fig. 3E, we disregarded the farfield sound, which was between −12 and −18 dB below nearfield sound at our measurement distance of r=0.21 m (Fig. 6B). The −18 dB estimate derived from assuming force is produced at n=1 (the fundamental frequency); the −12 dB estimate derives from assuming force is produced at n=2 (the 2nd harmonic). The data shown in Fig. 3C show the two harmonics are of similar magnitude in z, implying the actual amplitude of the farfield was between −12 and −18 dB. Accounting for it would require attention to phase differences between the nearfield and farfield. Inclusion of farfield sound would reduce the 2 dB discrepancy between model and measurement in the z direction. Other subtle discrepancies between model and measurement in Fig. 3E might arise from the basis for Song et al.’s (2014) computational fluid dynamics model of a hovering hummingbird, which was taken from a male Ruby-throated hummingbird, whereas we recorded wing sound from a female Costa's hummingbird. There are pervasive differences between the sexes in hummingbird flight: male hummingbirds have substantially higher wingbeat frequencies and shorter wings than females. That is, our assumption that our hummingbird was a scaled version of theirs might be problematic, as the role of wing rotation in force production is likely greater in animals with increased wingbeat frequencies. A better test of the model would be to simultaneously record kinematics for a CFD model, and record wing tone from the same animal.

The crucial assumption of our model is that the amplitude of P(i,t) around ⁠, which we called P_hov, is proportional to total force (mg in the case of hovering) and, hence, the magnitude of P_hov∝mgr⁻² (nearfield) or P_hov∝fmgr⁻¹ (farfield) (Fig. 4C). This approximation will underestimate mg if some portion of lift is produced continuously (rather than episodically) (Fig. 4C, case 2), or overestimate mg if the animal produces substantial negative lift at some point during the wingbeat (Fig. 4C, case 3). Although neither of these conditions seems likely to be general to hovering, case 2 is likely representative of sound on the animal's z-axis during forward flight, because the mean air flow across the wings is sufficient to produce some fraction of lift continuously.

The model presented here makes quantitative predictions for hovering (Fig. 6). While large birds such as albatrosses, eagles or barn owls do not hover, they do make Gutin sound in the modes of flight they do engage in, such as takeoff (Boonman et al., 2018) or forward flight (Boonman et al., 2020; LePiane and Clark, 2020). Eqn 1 can be applied to other scenarios. Consider an owl gliding at a constant velocity towards a mouse. When gliding, the owl does not hear its own Gutin sound, as its ears move in the same frame of reference as its wings. But the mouse is exposed to rising pressure caused by the owl's approach. When gliding at constant velocity, F(i,t) is time invariant, δF_i/δt, δQ/δt and δM_i/δt are zero, and if it glides straight at the mouse, M=M_r, causing Eqn 1 to simplify to: P(i,t)=F_i/[4π(1−M_r)²r²]. The increase in pressure at the mouse caused by the owl's approach is low in magnitude, as F_i in the direction parallel to the owl's forward trajectory corresponds to drag, which is lower in magnitude than lift. The sound is also very low frequency, as the time course of change of pressure is associated with only the decreasing r as the owl approaches.

Animals use sound in communication (wing tone may be selected for) and in predator–prey interactions (wing tone may be selected against). The sound modeled here is an intrinsic byproduct of producing oscillating aerodynamic force with flapping wings. Wing sounds have been evolutionarily co-opted for communication many times in birds (Clark, 2016, 2018), but it is usually not the Gutin sound itself that plays a communication role, presumably because it is low frequency. In insects, mosquitoes and midges communicate with Gutin sound (Cator et al., 2009; de Silva et al., 2015) in flight. As Seo et al. (2019) point out, harmonic frequencies propagate further than the fundamental frequency; thus, species selected to use Gutin sound for communication may be selected to adopt wing morphologies and kinematics that accentuate harmonic frequencies. In addition to mosquito wing sounds, wing-generated sounds that appear to be Gutin sound serve communication purposes in Drosophila courtship (Spieth, 1974) or the waggle dance of bees (Michelsen et al., 1986) (Table S1), and there may also be plants that respond to insect wing sounds (Veits et al., 2019).

Regarding hunting, insects such as mosquitoes that hunt on the wing will inevitably give off sounds to which their prey may be sensitive. For instance, caterpillars respond to tones between 50 and 900 Hz with anti-predator behaviors, the frequency range corresponding to the wingbeat frequency of insect predators and parasitoids of caterpillars, such as flies and wasps (Taylor and Yack, 2019, and references therein). Taylor and Yack (2019) present data suggesting monarch butterfly (Danaus plexippus) caterpillars are sensitive specifically to near-field sound. Our Eqn 4 provides an estimate of the nearfield sound a parasitoid or predator makes when approaching possible prey, based on the mass and wingbeat frequency of the predator. Detection of low-frequency nearfield sound (corresponding to insect wing hum) is widespread in invertebrates (Taylor and Yack, 2019). For instance, orb-spiders drop out of their web at the approach of a hermit hummingbird that is hunting them, reportedly in response to the wing hum (Stiles, 1995).

Owls and other nocturnal birds (Caprimulgiformes) seem to have evolved mechanisms to reduce aspects of their flight sounds (Boonman et al., 2018; Graham, 1934). As the Gutin sound modeled here is an inevitable result of the generation of oscillating aerodynamic force with flapping wings, the only way an animal may reduce its Gutin sound signature is by modifying the time course of the aerodynamic forces it produces with its wings, such as by gliding rather than flapping (as in the thought example presented above). As most birds have wingbeat frequencies below 20 Hz, the Gutin sounds they produce may often be at frequencies too low to be audible to prey. But this is not certain. For example, while most rodents have limited hearing below 1 kHz, desert rodents have independently evolved sensitive low-frequency hearing multiple times (Mason, 2016; Webster and Webster, 1980). Manipulating a kangaroo rat's (Dipodomys) hearing impairs its ability to evade an owl strike (Webster, 1962; Webster and Webster, 1971). Moreover, spiny mice (Acomys cahirinus) behaviorally react to the wing flaps of an owl taking off (Boonman et al., 2018; Ilany and Eilam, 2008). Whether these rodents detect owls by sensing the Gutin sound of their wings, rather than other higher-frequency sources of sound that an attacking owl makes, remains unclear.

We thank Jasmine Mitchell and Irini Saad for assistance with data collection, Al Muth for access to Boyd Deep Canyon, and Krista LePiane and Arjan Boonman for sharing data.