Ballistic properties in ancient Egyptian arrows

P.H. Blyth, Ph. D.

This article was first published in the Journal of the Society of Archer-Antiquaries, volume 23, 1980.
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Ancient Egyptian arrows, which have been recovered in large numbers, and often in a very good condition, present something of a puzzle. Thy are very light (usually about 0.35-0.5 oz., 10-15 g) rather long (usually between 30 and 33 in., 0.76-0.84 m), and made of a slender reed giving a large deflection (outside diameter usually 9/32 - 5/16 in., 7-8 mm). Using a modern 'spine' table, they would seem suited to a bow with a pull of only 15 lb. (67 N) at full draw. On the other hand the bows with which they were found - both the doubly convex self longbows, and the composites - look as though they drew about 40 lb. (178 N); the ability of the arrows to make deep and lethal wounds is amply attested (e.g., Winlock, 1945); and they remained the standard equipment of the Egyptian army for several millenia, more or less unchanged except for the warhead. Are we therefore to suppose that some special skill allowed the archers to shoot them with bows which would by modern standards be too powerful, or that the bows were by modern standards very inefficient, and that the arrows only achieved a reasonable range because of their low weight, and reasonable penetrating power because of their small heads?

That the archers may have had a special technique may be surmised from ethnological parallels. It is in general typical of primitive technologies that they compensate for a low level of design by great skill in use, whereas high technologies do the reverse, and archery is no exception. Saxton Pope (1962) found that most Red Indian arrows were very soft and that he hinself could not shoot with the equipment made by Ishi, the Yaqui Indian. 'Spine', as was shown by Rheingans and Nagelr (1937), has to do with the lateral vibration of the arrow, ensuring that it passes the bow-stave without hitting against it; the amplitude of the vibration (though not its frequency) is considerably affected by the method of release, and even within our own tradition, arrows must to some extent be matched to the archer as well as to the bow; it is therefore quite credible that primitive archers should to some extent be able to compensate for low vibration frequencies, though they probably also put up with lower accuracy, shooting either en masse, in war, or at short range, when hunting. (Ishi himself performed rather poorly at target shooting.) If so, it follows that we must find some other method than direct comparison with modern practice, if we wish to assess the performance of these arrows; we cannot follow directly in the footsteps of Pratt and Hardy (1977, and in Hardy, 1976, 198-204) in their treatment of the Westminster arrow. In this article I shall propose a method which makes fewer assumptions (though its results are less definite), and apply it to the measurements of a group of arrows in the Pitt Rivers Museum in Oxford. It will appear that the energy of the arrows must indeed have been very low, and the bows inefficient, though the entire system of bow, arrow, and arrowhead was quite well matched.

Minimum Spine

Common sense suggests that there must be some absolute lower limit to the stiffness of an arrow, in relation to its weight or mass, below which it will not leave the bow at all, but collapse under its own inertia. In practice, Mr. Hardy has described to me how a flight arrow designed for a 40 lb. bow seemed to explode when shot in an 80 lb. bow, and I found that a light arrow will fly out sideways from the bow when fitted with a heavy head. This is a type of collapse which no amount of skill in shooting can prevent, since the arrow becomes unstable even under a force aligned exactly along its axis once that force exceeds a critical value, Fcrit, which can be predicted by Euler's formula for the buckling of a thin strut. That formula, which is given in an appendix below, can be rearranged (as explained there) to express the critical load in terms of measurements usually made by archers as

Fcrit = ((S+H)/(S+2H))*(C/(yL^2))

where S is the mass of the shaft, H the mass of the head, y is the spine of the shaft, L is its length, and C is a constant. (If y is expressed in standard G.N.A.S. units (deflection, measured in hundreths of an inch, when the shaft is supported at 26 inch centres and loaded in the middle with 1.5 lb.), C will have a value of C=1,000,000 lbf when L is measured in inches, and C=2,880 N when L is measured in metres. If the spine is measured using a 2 lb. load, these values must be increased by one-third. S and H can, of course, be measured in any units convenient).

Thus an arrow 26 in. (0.63 m) long with a spine of 40 G.N.A.S. units and a head of negligible mass will have a critical buckling load

Fcrit = (1,000,000/(40 x 676)) = 36 lbf (163 N)

while if we add a head equal in mass to one-quarter of the shaft, Fcrit will become 36 x 1.25/1.5 = 30 lbf (135 N).

At first sight, these seem very low values. A standard table matches an arrow with this amount of spine with a bow drawing 60 lbf (270 N) and it might look as though the arrow should buckle as soon as it is released.

However, the 'weight' of the bow is, of course, the force on the fingers at full draw; it is the force they exert in keeping the string there, and if the arrow itself exerted such a force when the fingers were withdrawn, it would not move at all. Since action and reaction are equal, the string can only exert a force on the arrow equal to that exerted by the arrow on the string, and it will be proportional to the mass of the arrow multiplied by its acceleration. If an arrow is to reach a velocity of around 165 ft./sec. (50 m/s) within about two feet (0.6 m) its average acceleration over that distance must be around 200 G, implying an average force on a 1 oz. (28 g) arrow of 200 ozf or 12.5 lbf (52 N). This makes the arrow in our example fairly safe, but since the acceleration is certain not to be uniform we need to consider the peak load, and the shape of the acceleration curve, which will vary in bows of different design.

A few curves showing the acceleration of arrows in a longbow were published by Hickman (1929). They show a high peak soon after the release, about twice the average, followed by a long plateau around the average for the total distance. Figures are as follows:

Draw weight of bow Weight of arrow Efficiency of shot
(lbf) (N) (gr) (g) (%)
3223015 43
32 36524 49

Acceleration Force on arrow
average max average max
(G)(G) (lbf)(N) (lbf)(N)
200 3786.4 (28) 12.0 (53)
145 340 7.5 (34) 17.7 7.9

Both the bow (lemonwood, with fibrebacking) and the arrows were rather light, and the efficiency of the combination was rather low; however, pending further experiments these figures provide a starting point for an informed guess as to what may happen elsewhere, in the relationship between the peak force and the average acceleration which governs the maximum velocity of the arrow. As we should expect, the peak force on the lighter arrow is lower, but the acceleration appears more uniform; in fact, while the difference between the peak loads on the two arrows is closely proportional to the difference in their mass, the average load differs much less. This is quite credible when we consider that most of the energy during the later part of the shot, at least in a longbow, comes from the kinetic energy stored in the arms of the bow, and that they will have been slowed down by the heavier arrow more than by the light one. Light arrows may in general have a flattish acceleration curve when shot from massive bows, even when the bows are fairly heavily 'stacked' so that there is very little elastic force in the arms during the later part of the shot (as would seem to have been the case with the doubly convex Egyptian longbow). However, other things being equal, such 'stacking' must tend to produce a higher peak early in the shot, while conversely a reflex or compound bow will give smoother acceleration. If we guess that the peak usually has a value between three times the average (in a heavily 'stacked' longbow) and 1.5 times the average (in a reflex or compound bow with a light arrow), we may not be far wrong.

This relationship between the peak force and the average is, for archaeological purposes much the most important, since we can use it to compute a limit to the maximum velocity attainable in a bow of each type with an arrow of known mass, length, and bending strength. To go further, and relate the peak force to the draw-weight of the bow, would involve too many variables to make it worth while. It is interesting to note, however, that Hickman's arrows, if they were suitably matched to the bow, will have had a safety factor of around 2, like that of the imaginary example, and that this is in accordance with Hardy's observation.