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A single universal equation can approximate the frequency of wingbeats and finstrokes made by birds, insects, bats and whales, despite their different body sizes and wing shapes, Jens Høygaard Jensen and colleagues at Roskilde University in Denmark report in a new study published in PLUS ONE on June 5.

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

The researchers used size analysis to calculate an equation that describes the frequency of wing beats of flying birds, insects and bats and fin beats 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 mass divided by the area of ​​their wings. They tested the equation’s accuracy by plotting its predictions against published data on the wingbeat frequencies of 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 strokes in penguins and several species of whales, including humpbacks and northern bottlenose dolphins.

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

Finally, they calculated 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 study shows that despite vast physical differences, animals as diverse as butterflies and bats have evolved a relatively constant relationship between body mass, wing area and wingbeat frequency.

The researchers note that they did not find publications with all the necessary information for swimming animals; data from different publications were pooled together to make comparisons, and in some cases animal densities were estimated based on other information.

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

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

The authors add: “A difference of almost a factor of 10,000 in wing/fin beat frequency, data for 414 animals from blue whales to mosquitoes fall on the same line. As physicists, we were surprised to see how well our simple prediction of the wingbeat formula worked for such a diverse collection of animals.”