The beat frequency of birds, insects, bats and whales predicted using only body mass and wing area

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

The ability to fly has evolved independently in many different animal groups. To minimize the energy required to fly, 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.

Researchers used dimensional analysis to calculate an equation describing the frequency of the wing beats of flying birds, insects and bats, and the fin beats of diving animals, including penguins and whales.

They discovered that flying and diving animals beat their wings or fins at a frequency proportional to the square root of their body weight divided by their wing area. 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 the frequency of fin blows for penguins and several species of whales, including humpback whales and northern bottlenose dolphins.

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

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

The research shows that despite major physical differences, animals such as butterflies and bats have developed a relatively constant relationship between body weight, wing area and wing beat frequency.

The researchers note that they have not found any publications with all the necessary information for swimming animals; data from different publications were pooled to make comparisons, and in some cases animal density was estimated from other information.

Furthermore, extremely small animals – smaller than any discovered to date – would likely not fit into this equation, because the physics of fluid dynamics changes on such a small scale. This could have consequences for flying nanobots in the future.

The authors say the equation is the simplest mathematical statement that accurately describes wing beats and fin beats in the animal kingdom.

The authors add: ‘The data from 414 animals, from the blue whale to mosquitoes, differ in wing/fin beat frequency by almost a factor of 10,000. As physicists, we were surprised to see how well our simple prediction of the Wing-beat formula works for such a diverse collection of animals.”