Flapping frequency of birds, insects, bats and whales predicted by body mass and wing area alone

In a new study published in 1999, Jens Højgaard Jensen and colleagues from Roskilde University in Denmark report a single universal equation that can accurately approximate the frequency of wing and fin beats made by birds, insects, bats and whales, despite their different body sizes and wing shapes. PLOS ONE on June 5.

The ability to fly evolved independently in many different animal groups. To minimize the energy required for flight, biologists expect that the frequency at which animals flap their wings should be determined by the wing’s natural resonant frequency. However, finding a universal mathematical description of flapping flight has proven difficult.

The researchers used dimensional analysis to calculate an equation that describes the wingbeat frequency of flying birds, insects and bats and the fin strokes of diving animals, including penguins and whales.

They found that flying and diving animals beat their wings or fins at a frequency that is proportional to the square root of their body weight divided by the area of ​​their wings. They tested the equation’s accuracy by plotting its predictions against published data on wingbeat frequencies for bees, moths, dragonflies, beetles, mosquitoes, bats and birds ranging in size from hummingbirds to swans.

The researchers also compared the equation’s predictions with published data on fin stroke frequency for penguins and several whale species, including humpback whales and northern whales.

The relationship between body mass, wing area and wingbeat frequency shows little difference between flying and diving animals, despite vast differences in their body size, wing shape and evolutionary history, they found.

Finally, they estimated that the extinct pterosaur (Quetzalcoatlus northropi)—the largest known flying animal—beat its 10-square-meter wings at a frequency of 0.7 hertz.

The study shows that despite vast physical differences, animals as different as butterflies and bats have evolved a relatively constant relationship between body mass, wing area and wing beat frequency.

The researchers note that they did not find publications with all the information required for swimming animals; data from different publications were pooled to allow for comparison, and in some cases animal density was estimated from other information.

Furthermore, extremely small animals—smaller than any discovered so far—would probably not fit the equation because the physics of fluid dynamics changes on such a small scale. This could have implications for flying nanobots in the future.

The authors say the equation is the simplest mathematical explanation that accurately describes wing beats and fin strokes across the animal kingdom.

The authors add: “Differing in wing/fin beat frequency by almost a factor of 10,000, the data for 414 animals from blue whales to mosquitoes fall into the same line. As physicists, we were surprised at how well our simple prediction of the Wing Beat Formula works for such a diverse collection of animals.”