The arrow in a crosswind: Park 2020 study

The paper calculates how far a side wind moves an arrow in flight, and for the first time in the author's models it includes the flexing of the shaft. With a steady 3 metres per second wind, a light breeze, blowing from the right at 90 degrees for the whole flight, the drift is large even with the best arrows: for recurve at 70 metres it ranges from 245 to 261 mm for X10 600 shafts, depending on point weight, more than four scoring rings on the 122 cm face, each 61 mm wide; for compound at 50 metres it is 75 mm for X10 and Protour 380, nearly two 40 mm rings.

The drift has two parts. One comes from the arrow turning into the apparent wind as it aligns itself with the airflow; the other from the sideways component of aerodynamic drag, which is by far the larger, about ten times the first. In the case with 10 mm of initial flex, for example, an X10 600 with a 140-grain point drifts 20 mm from alignment and 225 mm from drag. Normal shaft flex changes the total little, whereas an initial rotation of the arrow can change it considerably.

Shaft diameter is the biggest factor: drift grows roughly linearly with diameter. At 70 metres it goes from 253 mm for the X10 (4.92 mm average diameter, 120-grain point) to 297 for the Carbon One, 311 for the ACE and 441 for the aluminium X7; at 50 metres from 75 mm for X10 and Protour to 87 for Carbon One and ACE, 124 for the Light Speed and 160 for the Fat Boy. For a given diameter a heavier arrow drifts less: going from a 100 to a 140-grain point takes the X10 600 from 261 to 245 mm. A point shaped to keep the airflow laminar over the front of the shaft would take off a further 8.3% at 70 metres and 4.0% at 50. According to the author, drift grows with the square of the distance.

The arrow is described as a flexible, inextensible beam whose diameter, stiffness and mass vary along its length, divided into 16 segments; the equations are solved step by step with a finite difference method, at intervals of 0.025 ms. Drag and lift are calculated separately for the point, shaft, fletchings and nock, each with its own local angle of attack, which varies along the arrow because the shaft flexes. The trajectory is recalculated at every step, and the launch angle is adjusted by trial until the arrow arrives at the right height on the target.

The bows are two of the author's own, measured by him. The recurve is a Hoyt Formula X at 164.9 N (37.1 pounds) at a 700 mm draw length, shooting a 21.30 g X10 600 at 57.2 metres per second; the compound is a Hoyt Pro Comp Elite with a peak draw force of 237.9 N (53.5 pounds), shooting a 24.86 g X10 Protour 380 at 78.9 metres per second. These measurements give the speed of the other arrows, all Easton, 710 mm long, with the points typical for each model. For recurve the author used an earlier method of his to pick arrows that would leave the bow behaving close to optimally at the same draw weight, with small adjustments to the pressure button; for compound he picked arrows of similar spine.

Five cases were studied, with and without wind, with an initial angle of attack of one degree, an initial rotation of 10 degrees per second or 10 mm of flex. The model is compared with another research group's model on the same arrows: 267 against 260 mm of drift for the X10 600, and 411 against 360 for the ACE 1000. The frequency at which the arrow oscillates about its path, about 3.7 Hz, was checked by shooting an arrow through sheets of paper at different distances.

In this paper drift is not measured in the field: it is calculated, and the comparison is with another model, which agrees well for the X10 and less well for the ACE; the rate at which the oscillation dies away also differs a great deal between the two models, for both arrows. The only experimental check concerns the oscillation frequency, read from paper sheets. The wind is constant, uniform over the whole 70 metres and exactly sideways, whereas in competition it comes in gusts, changes along the range and also moves the archer and the aim, which do not enter here.

There are two bows, each at a single draw weight, and the 37-pound recurve is lighter than many elite archers use: with a stronger bow and a faster arrow the millimetres would be different. Arrows from other makers are assumed to behave like Easton arrows of the same diameter and mass, which is the author's assumption. The gain from a point that keeps the flow laminar comes from a model and an earlier study, and is estimated with the point's angle of attack staying mostly below 3 degrees; for stronger winds the author does not claim it will hold.

The first point is one of scale: with a 3 metres per second breeze, which many would not even call wind, a recurve arrow at 70 metres moves by about a quarter of a metre. Anyone shooting outdoors corrects for this anyway, by aiming off, canting the bow slightly or adjusting the sight; the study shows how much the choice of arrow weighs on that correction, everything else being equal. Between an X10 and a large-diameter aluminium arrow, at 70 metres, the difference is nearly 19 cm; between the common carbon arrows it is a few centimetres, in that same wind.

Diameter matters more than mass. A heavier point helps a little (16 mm at 70 metres going from 100 to 140 grains), but for a recurve archer changing point weight means changing how the arrow behaves dynamically, and therefore the right spine and the tuning: in the study the arrows were chosen specifically to leave the author's bow cleanly. According to the author, today's thinnest shafts are already close to the limit, because below a certain internal diameter there is not enough room left for a sound point; he therefore expects no further gains from reducing diameter. The square-of-distance rule, finally, means that at half the distance the same breeze moves the arrow by about a quarter as much.