Ancient bows and mathematical models: Kooi 1997

The authors use a mathematical model of the bow to answer a question from archaeology: what advantage did a composite bow with stiff ends actually give over a straight wooden bow? The answer they reach is that at equal draw weight, draw length and limb mass, with the same arrow and the same string, initial velocity changes little from one type to another. The dimensionless coefficient proportional to velocity comes to 2.01 for a Klopsteg flatbow, 1.92 for the angular bow of Egypt and Assyria, 1.94 for the Persian bow, 1.99 for the Turkish flight bow, 2.08 for an excessive-recurve bow and 2.09 for the modern recurve used earlier to validate the model.

The reason lies in the way the bow's two virtues trade off against each other. Bows with stiff recurved ends store far more energy at full draw (the static quality coefficient is 0.432 for the Persian bow and 0.491 for the Turkish one, against 0.407 for the flatbow), but they pass a smaller share of it to the arrow, because those ends are heavy and, at the end of the stroke, decelerate sharply and leave energy in the vibrating limbs and string: efficiency falls to 0.668 for the Persian bow and 0.619 for the Turkish one, against 0.765 for the flatbow. The excessive-recurve bow takes the case to its limit, with a static quality coefficient of 0.810 and an efficiency of 0.417. The modern recurve is the only one that holds both together, at 0.434 and 0.770, and gives the highest velocity, though by a small margin. Over the record distances of flight shooting, then, the Turkish bow does not win through the shape of its ends: it wins because horn and sinew store far more energy per unit of mass than wood (about 1,500 and 2,500 newton metres per kilogram against 1,100 for yew), which allows a very light and very strong bow, 690 newtons of draw weight for 350 grams of limbs and a length of 1.14 metres in the example tabulated.

The model treats the bow as two equal elastic limbs, symmetric about the line of aim and separated by a rigid grip, and discretises them into rigid elements with concentrated masses joined by elastic hinges, with the Bernoulli-Euler equation for curvature and Newton's second law for the motion. The string is taken as inextensible and its mass is shared out, one third with the arrow and two thirds to the limb tips, except in the recurve cases, where only two springs are used. The calculation requires every design parameter: the length of the limbs and of the grip, the unstrung shape, the cross-section at every point, the elastic properties and specific mass of the materials, the shape and mass of the stiff ends, the brace height, the draw length, the mass and elasticity of the string and the mass of the arrow.

The comparison runs on three dimensionless coefficients: the static quality coefficient, that is the recoverable energy stored in going from braced to full draw divided by draw weight times draw length; efficiency, that is the share of that energy which reaches the arrow as kinetic energy; and a third coefficient proportional to initial velocity. To keep the comparison honest, the six bows (Klopsteg flatbow, angular, Persian, Turkish, excessive-recurve and modern recurve) are computed at equal draw weight, draw length, limb mass, arrow mass and string mass.

The validation comes from two earlier pieces of work. The draw weight computed for a replica of the yew longbows recovered from the Mary Rose, measured parameter by parameter, matches the measured value to within 1%. For the dynamic behaviour the comparison is with measured arrow velocities from a modern bow of maple and fibreglass: there the predicted draw weight came out too high and the authors introduced a knock-down factor on the flexural rigidity of the limbs, while the measured efficiency came out a few percentage points below the computed one, which they attribute to internal and external damping, neglected by the model.

This is a calculation rather than a measurement. Nobody shot anything, and the validation covers two bows, one statically and one dynamically, rather than the four historical types that are then compared: for those, design shapes are simulated from the archaeological literature and, for the Turkish bow, from a nineteenth-century description. The model neglects internal damping in the materials and damping in the air, and in the one dynamic test available the measured efficiency was a few points below the computed one; the computed draw weight was too high and a knock-down factor on rigidity was introduced to make the figures match. One detail bears on the comparison: the modern recurve in the table has a string about twice as stiff as the others, which the authors state openly.

The bow is symmetric by assumption, so the model does not describe bows that bend through the grip, as many Native American bows do, nor the Japanese bow, whose grip sits a third of the way along and which bends asymmetrically. Comparing at equal draw weight, draw length and limb mass is the choice that makes the comparison possible, but it sets aside precisely what separated historical bows from each other, namely how much draw weight could be had from how much material: the advantage of the Turkish bow comes from the table of materials, not from the three coefficients. The computed initial velocity does not turn into distance: range also depends on the launch angle and on air drag, neither of which is in the model. Finally the model says nothing about accuracy, nor about how smoothly a bow shoots, which the authors name as an important criterion for target archery and admit they cannot reduce to a single number.

For an archer today this study works above all as an antidote to a way of reasoning. The reasoning is the one that takes a feature of the equipment, say the shape of the ends or the amount of energy stored, and reads it as an advantage. Here the calculation shows that the bow's two qualities are paid for against each other: storing more energy calls for heavier limbs, heavier limbs give back a smaller share of that energy, and the net result, with everything else held equal, is a nearly identical velocity for bows that look as though they belong to different ages. The modern recurve gains something, but only a little, and the margin comes from the materials more than from the design.

It also makes clear why equipment tests cannot be compared unless everything else is held fixed. The authors can say anything at all about the six bows only because they compute them at equal draw weight, draw length, limb mass, arrow mass and string mass: change one of those terms and the figures move and the comparison collapses. That holds for a mathematical model from 1997 just as it does for a comparison between two sets of limbs on a shooting field today. In the registry the source is a simulation: it counts as context on equipment, not as evidence about what an archer measures on the target.