Three scientists from Roskilde University in Denmark have found that almost all animals that fly in the air and many that swim in the water have evolved to flap their wings at (or on) a frequency given by a simple formula.

This formula relates the frequency at which a winged creature flaps its wings to hover in the air (or stay submerged) to the creature's mass and wing size.

The researchers also found that the formula applies to a wide variety of lifeforms, including insects, birds, bats, penguins, whales and a robotic bird called an ornithopter — the newest version of a dragonfly-like airplane. *Dune* Movies.

**What is the formula?**

The formula is straightforward: *F* ∝ √m/a. *F* Here is the wing flap frequency, m is the mass of the animal flying in the air, and A is the wing area (∝ means 'is proportional to').

When the researchers calculated the number of different animals, birds and insects √m/A and plotted it on the x-axis and their respective frequencies on the y-axis, they got an almost straight line (see below). The black line is the relationship based on the formula predicted by their model – an almost perfect fit.

A plot showing the scaling relationship between *F* and √m/A. | Photo credit: PLoS ONE 19(6): e0303834.

The Roskilde team's results were published in the journal *one more* On June 5th.

**How did they get the formula?**

People have tried to find some kind of unifying formula through the wing flapping frequencies of flying animals, birds and insects before. In most of these attempts, scientists arrived at correlations between wing flapping frequency and mass and wing area and mass based on measuring these numbers in the wild. The results were as follows: *F* ∝ m^{-0.43}Which scientists have not been able to reach based only on the knowledge of physics, without empirical data.

In their paper, the Roskilde team wrote that its members wanted to try the former approach: deriving a theoretical equation and then checking whether it matches the equation found in the wild. As they put it: “Can one arrive at a reliable prediction for the wing-beat frequency of a flying animal from physics-based arguments alone, i.e. without resorting to empirical correlations?”

He started with the equation for Newton's second law for an animal trying to stay in the air by flapping its wings: *F* = m*A*. The *A* here actually *Yes*is the downward acceleration due to gravity, and m is the mass of the animal. *F* This is the force the animal has to generate by flapping its wings. From here he worked forward, taking into account the speed of the air pushed downward by each wing stroke, the speed of the air flow around the wings, and the density of the atmosphere.

In doing so, he found a number of quantities – notably the size of the wings and the angle at which they flapped – that had no units, that is, they were dimensionless. He collapsed the numbers representing these units into a single constant called *C*; its exact value depends on empirical observations. Thus, he had his equation:

*F* Here is the wing beat frequency, m is the mass of the animal, _{Air} The density of the atmosphere, and the shape of the wing. And that's it.

**What does it do C Catch?**

Matt Wilkinson, director of studies in natural sciences at the University of Cambridge, said the equation's proportionality constant, *C*More information may also be available from this.

They told *Physics World* A bird's flight is most efficient when its wings beat at the bird's specific resonance frequency, but if the bird is heavy beyond a certain limit, this frequency will not be high enough to support its mass, so its flight will necessarily be less efficient.

He noted that, given the variable components of the equation, the dependence on size should be hidden in the constants. *C*“Despite the huge differences in wing shape and flight dynamics, understanding this is what will provide real insight,” he told the magazine.

**Does this formula apply to fish as well?**

Mathematics is a language for describing the natural universe and this equation represents a universal law that tells us why flying birds and insects fly the way they do, what a bird or insect that evolves in the future might look like, and what winged robots should look like if they want to fly in the air.

The proportional relationship that results when these adjustments are removed is, *F* ∝ √m/a.

Interestingly, their equation “also gives a prescription for the fin/fluke frequency of swimming animals because positively buoyant diving animals must constantly move water upward to stay submerged,” the researchers wrote in their paper. This, of course, is only true for animals that have no means of adjusting buoyancy, which does not include fish with swim bladders. Calculating the fin/fluke frequency [the equation] This requires two modifications, replacing the density of air with the density of water and the animal mass with buoyancy-corrected mass.”

**Where does this formula apply?**

The researchers tested their equation with data on birds, insects, etc. presented in older published studies, where they found “176 different insect data points (including bees, moths, dragonflies, beetles, and mosquitoes), 212 different bird data points (ranging from hummingbirds to swans), and 25 bat data points.”

The equation has a rearrangement in which its left side *is equal to* Its right-hand side, which assumes certain physical conditions. For example, you may remember learning about the Reynolds number (Re) in high school: at a low value of Re, the flow of a fluid will be streamlined; if the value is high, the flow is said to be turbulent. The value of Re depends on the density, speed, viscosity and length scale of the fluid. At high Re, the density of the fluid matters more than its viscosity to an animal trying to 'fly' through it – and this is the case with insects, birds, bats, etc., whose flapping frequencies the equation captures.

The authors write in their paper that this would need to be modified at low or very low Re, where fluid viscosity matters more than density. For example, they were able to find that “for animals flying at very low Reynolds numbers, *F* ∝ m/A replaces the above.”

They were also able to determine that the equation could not be simplified further unless the animal densities differed by an order of magnitude (i.e. a factor of 10) or more.