For dolphins, sharks, and bony fish moving at their preferred cruising speed, the ratio of tail frequency and amplitude to forward speed is constrained to a narrow but efficient range of values. This dimensionless ratio is the Strouhal number, and evolution seems to favor efficient swimming motion with a Strouhal number in the range of 0.2–0.4. In the October 16th issue of Nature, Graham K. Taylor et al. show that Strouhal numbers for birds, bats, and insects flying at cruising speed also seem to be constrained to a similar range of values.1

I was surprised to read that the same range of values applied across creatures of wildly different size, mass, and flight patterns, as well as across species that evolved the ability to fly independently. In particular, although the authors of the Nature article provide a chart of Strouhal numbers for flapping flight and swimming along with their statistical analysis, I wanted to see what ratios of amplitude, frequency, and speed actually look like in winged flight, what the Strouhal number actually represents, and why it is dimensionless. The Strouhal number For an animal or insect in flight, the Strouhal number is determined by the frequency (f ) of wing strokes, multiplied by the amplitude (A) of the wing, divided by the animal’s forward speed (U) through the air. The Strouhal number is equal to f A/U, or : f A

U Before discussing what this ratio represents, let’s look at each of the three components: frequency, amplitude, and speed. Frequency For winged flight, frequency (f ) is simply the number of wing beats per second. A complete wing beat has two parts, a downward flapping stroke for power and lift, and an upward stroke for recovery.

For example, when the Common Sheathtail Bat (Taphozous georgianus) flies at cruising speed, it beats its wings an average of 8 times each second, with one stroke and one recovery every .125 seconds.

The Zebra Finch (Taeniopygia guttata), cruising at a much higher frequency of 26.9 wing beats per second, beats its wings almost three and a half times within the same amount of time.

Although beats per second is an appropriate unit for measuring winged flight, frequency can also be measured more generally, in Hertz (Hz). Since 1 Hz = 1 cycle per second, the Common Sheathtail Bat cruises at a frequency of 8 Hz, and the Zebra Finch cruises at 26.9 Hz. Note that these diagrams assume direct flight with an even, symmetrical stroke and recovery. In the real world, many animals use an asymmetrical wing beat with a slower downstroke and a faster upstroke, and some smaller birds use intermediate flight patterns that alternate flapping with bounding or gliding motions.2,3

Amplitude

Also known as wingtip excursion, the amplitude (A) of the stroke is the vertical distance traveled by the tip of the wing during the flapping stroke. Different species use different angles of attack when flying, so a correct display of amplitude must either measure perpendicular to the angle of attack, or normalize the angle by rotating each animal to the vertical.

An animal’s body also naturally rises or falls with every wing beat, creating an undulating motion that can obscure the actual movement of the wing relative to the body. To show wing movement accurately, a relative flight line is best, with the animal’s body maintaining an even height through the entire wing beat.

Assuming a symmetric flight motion, a normalized angle of attack, and a relative flight line, the wing tip descends in a smooth curve through the flapping stroke (red line), and rises through the recovery (gray line), tracing an even waveform.

Amplitude is simply the height of the flight waveform, so the Common Sheathtail Bat cruises with an amplitude of 26 cm, while the Zebra Finch cruises with an amplitude of 12 cm. Of course in the real world different species use different flapping motions and flight patterns, but simplified waveforms allow for easy comparison.