Arrow speed from sound: French 2007 study

Two engineers at Purdue University show that the speed of an arrow can be measured from the sound it makes passing over a microphone. It needs a point that whistles on a single note: as the arrow approaches the pitch rises, and as it moves away it falls, because of the Doppler effect, and the speed follows from the difference between the two frequencies. In the test recording the whistle tended towards 4,635 Hz on the way in and 3,310 Hz on the way out, which with a speed of sound of 338.3 m/s gives 56.42 m/s.

The main test used three microphones in a line between archer and target and two matching arrows with the same whistling point, one with parabolic feathers and one with untrimmed feathers, of the kind used for shooting birds. Fitting the curve of frequency against time, the arrow with parabolic feathers came out at 56.98, 56.08 and 56.08 m/s over the three microphones, the one with untrimmed feathers at 56.39, 53.95 and 53.64 m/s. From the loss of speed the authors estimate a drag coefficient about three times higher for the untrimmed feathers (9.4 against 3.1, in the paper's units). The same curve also gives the arrow's height above each microphone, between 0.47 and 0.17 m, and so, in principle, its trajectory.

The tests were run indoors, along the diagonal of a room, with about 13 metres between the shooting position and the target. The three microphones sat beneath the arrow's path; sound was recorded with a data acquisition system, and the authors point out that the sound recorder on an ordinary PC would also do. The analysis uses a time-frequency distribution (Choi-Williams) to follow the whistle's note moment by moment, then fits an equation linking frequency to time through speed, the arrow's height above the microphone, its starting position and the whistle's frequency with the arrow at rest; the table also gives a reference time for each microphone. The speed of sound is taken from the standard atmosphere, as conditions in the room were very close to it.

For air resistance the authors assume a force proportional to the square of speed and consider only the horizontal component of the motion. The paper does not say which bow the arrows were shot from, how much they weighed or how many shots were taken: the table gives one set of values for each arrow, which appears to be a single shot.

This is a four-page technical note, with one shot per type of arrow and no comparison with a chronograph: the accuracy the authors claim is not checked against a second instrument. Some of the numbers raise doubts. The whistle's frequency at rest, which the model treats as a constant of the point, varies from one microphone to the next (from 3,860 to 3,845 Hz for one arrow, from 3,635 to 3,606 for the other), and the arrow with parabolic feathers has the same speed at the second and third microphones, which makes the drag estimate built on that loss of speed fragile. The arrows' mass and the units of the drag coefficient are missing, so the two values serve only to compare the two arrows with each other.

The method works only with a point that gives a clean note: ordinary arrows make a broadband rushing noise that is no use for this measurement, and a whistling point changes the weight and aerodynamics of the arrow being measured.

For anyone who shoots, the study is mainly a technical curiosity. It shows that an arrow's speed, and the way it loses speed along its path, can be worked out from an audio recording, and that large untrimmed feathers slow an arrow far more than standard parabolic feathers: over about 5 metres the arrow with untrimmed feathers lost almost 3 m/s, the one with parabolic feathers about 1. That gives a sense of scale for why fletching matters, not a figure for choosing feathers.

To measure the speed of your own shots a chronograph remains much simpler, since this method needs a whistling point, a quiet room or field and signal analysis that few archers can do. The remark the paper starts from, that consistent speed from shot to shot helps accuracy, is the authors' statement: the study does not measure it.