Fig 1.
Correlations between wing area, wing-beat frequency, and mass of flying animals.
Literature data for insects (blue diamonds) [21, 29–31], birds (red circles) [12, 13, 15, 30, 32, 33], and bats (green triangles) [32, 34]. (a) Wing-beat frequency versus mass; (b) wing area versus mass. These data confirm the intuition that large flying animals have a low wing-beat frequency and large wings. (c+d) Tests of power-law expressions for the wing-beat frequency (same data as in a+b): (c) f versus 1/A; (d) f versus 1/m. In both cases significant correlations are observed, with the Pearson correlation coefficients 0.95 and 0.85, respectively, and vanishing p-values… The best-fit slopes are 0.42 and 0.30, corresponding to power-law expressions f ∝ m−0.43 and f ∝ A−0.31. These empirical findings cannot be justified from first principles, however.
Fig 2.
Wing-beat-frequency data for a variety of flying animals versus the square-root of the animal mass divided by the wing/fin area (same data as in Fig 1).
There is some scatter in the data, which is not surprising given that these are for quite different animals, but to a good approximation the data fall on the same line. The model prediction is the full black line of unity slope unity and the Pearson correlation coefficient is 0.95. Data for swimming animals, corrected for buoyancy and the difference in water and air density (see main text for details), have been included in the plot. The uncorrected data for swimming penguins (open squares) [24], large whales [43] (open purple triangles), and small whales [44] (open light purple triangles) clearly do not conform to the general trend for flying animals. However, with the appropriate corrections, the data points (filled symbols) fall on the continuation of the best fitted model prediction line of the flying animals. Photo attributions: Hummingbird, republished from wikimedia with no changes made under a CC BY license (https://creativecommons.org/licenses/by-sa/4.0/deed.en), with permission from Charles J. Sharp: https://commons.wikimedia.org/wiki/File:Cinnamon_hummingbird_(Amazilia_rutila)_in_flight_Los_Tarrales.jpg. Moth, republished from Flickr under a CC BY license (https://creativecommons.org/licenses/by/2.0/), with permission from Judy Gallagher: https://www.flickr.com/photos/52450054@N04/35499910606. Penguin, republished from Hans Ramløv under a CC BY license, with permission from Hans Ramløv, original copyright [1991]. Beetle, republished from Thorbjørn Ramløv under a CC BY license, with permission from Hans Ramløv, original copyright [2018]. Whale, retrieved from RawPixel at https://www.rawpixel.com/image/4022862/whale-originalpublic-domain-image-from-flickr. Seagulls, retrieved from Pexels at https://www.pexels.com/photo/white-seagulls-flying-over-the-ocean-54462/. Mosquito, retrieved from Pexels at https://www.pexels.com/photo/black-white-mosquito-86722/. Bumble bee, retrieved from Pexels at https://www.pexels.com/photo/bumble-bee-on-yellow-daisy-67560/. Dragonfly, retrieved from Pexels at https://www.pexels.com/photo/yellow-and-black-dragonfly-on-green-stem-during-daytime-56010/.
_in_flight_Los_Tarrales.jpg){kind=link}
Fig 3.
Testing a simpler expression for the wing-beat frequency data (same data in Fig 1).
The plot shows a test of Eq (7), which is derived from Eq (6) assuming similar shapes and movements for all flying animals. The data do not follow the predicted proportionality f ∝ A−1/4 indicated by the full line. Instead, the best fit (dashed line) yield f ∝ (A−1/4)1.68 = A−0.42, which is consistent with Fig 1(c).