There are similarities in the way a bird flies through the air and a dolphin swims through the water. And yet how do you compare the flap of a butterfly to the enormous movement of a blue whale? The best way, it turns out, is with their respective Strouhal Numbers.
It's easy to figure what's the fastest moving creature in the skies (or the seas). Just clock the distance that any given fish, whale, bird, bat, or bug moves over a certain amount of time. Divide distance by time. Done. But is whatever you've proclaimed the winner really the fastest mover? You might be able to outrun a butterfly, but considering its size, could it be moving its body faster relative to a bird or bat that would chase you down? Absolute speed may be easy to distinguish, but actually comparison is tougher to do.
The Strouhal Number is designed to look past differences. To calculate it, start out with 'f,' the flap rate (strokes/time), and multiply it with the length 'l' of the stroke (distance/stroke). Then divide them both by the speed, 'V,' of travel (distance/time). So we get fl/V. When dividing by something, you invert its units, so the overall unit equation is (strokes/time) x (distance/stroke) x (time/distance). Those who paid attention in junior high math know that you can cancel out like terms if they appear on both the top and the bottom of a problem. For every unit that appears on the top of that equation, there is a unit that appears on the bottom. All the units cancel out. This leaves fl/V, the Strouhal Number, as a number without units no matter what is being compared to what. Inches can be compared to feet. Seconds can be compared to hours. A sting ray, a parrot, that weird dumbo octopus that was recently discovered on the ocean floor, anything that gets around by flapping can be compared using this unitless number.
But why would anyone care to compare? Well, it turns out that the Strouhal Number, invented by Vincent Strouhal just as the 1800s turned into the 1900s, is a way to compare the different species in a way that fits their massively different techniques into a narrow range. Almost all flapping creatures fit into a Strouhal range of 0.20 to 0.40. Birds of prey tend to circle around the 0.24 radius while bats are up in the 0.26 range. Dolphins and whales are an impressive 0.28. Different species with different evolutionary histories, tend to cluster around different Strouhal Numbers. This one unitless number shows the whole of evolutionary history.
Strouhal Numbers can also be used to show exceptionally good swimmers within their own species. The average college swimmer will have a Strouhal Number of 0.36. But long distance swimming champion Shelley Taylor-Smith, during an open-water marathon, had a Strouhal Number of 1.5. In absolute speed, it's doubtful that she can swim as fast as a shark, but the Strouhal Number doesn't use absolute speed, and so it can show her exceptional talent. Although the number compresses most swimming stats, a champion clearly comes through.
Image: Nasser Akabab