The arrow has fared rather meagerly, in comparison with the bow, as regards mathematical and mechanical analysis. Some experimental work has thrown light on its behavior as it passes the bow. The flight of the arrow, with due regard for air resistance, has come in for more analytical study, but the experimental data are scant. There has, however, been evidence of the value of some of the theoretical work that has been done. Of this we shall treat more fully below.
In two articles, Klopsteg describes the results of an investigation made, during the winter of 1932-33, by means of highspeed photography. The reader who is interested should consult the original articles. In them are reproduced a number of "lightning flash" photographs, showing clearly how an arrow "snakes" or oscillates its way past the bow. Other photographs show the path of the string, in side and plan views as well as the motion of the bow hand during the shot. Deductions from a study of about 200 photographs are given in The Archery Review for January, 1934. Hickman, through contact with Electrical Research Products, Inc., has recently had access to the high speed motion picture camera developed by that company, and has thus screened several reels of him showing the arrow being discharged by a bow, both in the shooting machine and by band, in extremely slow motion. These remarkable pictures confirm in all respects the "stills" taken by the "lightning flash" method.
The oscillation of the arrow, so clearly shown in the pictures, is the clue to what constitutes "matching" of an arrow to a bow. In the first place, the arrow must be stiff enough so that it will not perceptibly buckle under the acceleration given it by the string. At the same time, it must have a period of oscillation which is properly timed with the passage of the arrow across the bow handle. This means that it should oscillate rapidly enough, after the oscillation has been initiated by the loose, to insure that the rear end of the shaft will be vibrating away from the bow while it is passing the latter. If it vibrates too slowly, the tail of the arrow may strike the arrow plate with considerable impact, and the arrow is thrown to the right of the line of aim. If the arrow is too stiff, the impulse between the foreshaft and the arrow plate may cause considerable leftward deflection.
All this leads to a consideration of the much used and little understood term, "spine". Incidentally, the effect of spine upon accuracy of flight is greatly diminished by supporting the bow off center towards the arrow plate, so that the force of the string will act as nearly as possible along the axis of the arrow. Most bows exert their force in a line to the right of the arrow, through the middle or even to the right of the middle, of the bow handle. This introduces a so-called component of the force to the right which has the effect of increasing the sidewise thrust of the foreshaft against the arrow plate. When such action is present, it causes increased lateral errors, and makes equality of spine in a set of arrows exceedingly important.
Aside from the discussion of spine by this writer the subject has been treated in a paper by Rodgers and another by Rheingans. One must distinguish between spine of an arrow, and the spine characteristics of wood from which arrows of equal spine may be made. It should be possible to measure the latter, and in this way select squares or dowels from which arrows of equal spine may be made. Rodgers and Rheingans have given formulas for doing this.