February 15 19 12] 



NATURE 



533 



<ieflection in it. It has been found that the time of contact 

 measured in this way for steel balls is exactly that pre- 

 dicted by theory, and it may be inferred that the theory 

 is correct in all its details, and that the pressure calcu- 

 lated by its aid corresponds with the facts. This method 

 was first used by Pouillet in 1845, and has recently been 

 brought to great perfection by Mr. J. E. Sears, who 

 showed, among other things, that the relation between 

 pressure and deformation of steel is almost exactly the 

 same when the pressure is applied for an excessively short 

 time, as in the case of impact, as it is when applied 

 steadily, as in a testing machine. The assumption that 

 this is the case lies, of course, at the root of the calcula- 

 tions, and its verification was therefore a matter of con- 

 siderable importance. 



When one billiard ball strikes another the effect of the 

 iblow is practically instantaneously transmitted to every 

 portion of the colliding balls, or, to speak more precisely, 

 the time taken to transmit the pressure is short compared 

 with the total time of contact. Except for the minute 

 relative displacement near the point of contact, the balls 

 move as a whole, every part having the same velocity at 

 each instance of time and coming to rest at the same 

 moment. In many cases of impact, however, and in those 

 possessing the most interest from a practical point of view, 

 this is by no means the case. We may consider, for 

 instance, the impact of an elongated lead rifle bullet against 

 a hard steel plate. Under the enormous pressures 

 developed lead flows almost like water, and in the absence 

 of lateral support it is as little capable of transmitting 

 those pressures. Thus, when the nose of the bullet strikes, 

 the metal thus brought into contact with the plate immedi- 

 ately flows out laterally, its forward motion being 

 ■destroyed ; but the hind parts of the bullet know nothing 

 of what has happened to the nose, because the pressure 

 cannot be transmitted to them, and they continue to travel 

 on with the original velocity until they in their turn come 

 up to the plate and have their momentum destroyed. The 

 process of stopping the bullet is complete when its tail 

 reaches the plate, and the time required is simply that 

 taken by the bullet to travel its own length. Thus a Lee- 

 Metford bullet is i^ inches in length, or, say, one-tenth of 

 a foot, and if moving at 1800 feet per second, which 

 is about the velocity given with a rifle, it would be 

 stopped in 1/18,000 of a second. The bullet weighs 

 approximately 003 lb., and possesses with this velocity 

 about 1-7 lb. second units of momentum. The force re- 

 quired to destroy this in 1/18,000 of a second is 18,000 

 multiplied by 17 lb., or, say, 15 tons. This acts over the 

 sectional area of the bullet, which is one-fourteenth of a 

 square inch, giving a pressure of about 210 tons per square 

 inch. This is the average pressure throughout the impact, 

 but the pressure is probably nearly constant. It is to be 

 noted that the pressure per square inch depends only upon 

 the velocity (varying as its square), and not upon the 

 length or diameter of the bullet. Increase in diameter 

 only alters the area over which the pressure is applied, and 

 increase in length the time during which it is applied. 



If for the bullet of lead we substitute one of hardened 

 steel which will not flow, the problem at once becomes 

 much more complicated. In order to reduce it to its 

 simplest terms, and to bring the theory into such a form 

 that it can be tested in the laboratory, we may suppose 

 that, instead of the bullet, we have a cylindrical steel rod, 

 say J inch in diameter by 10 inches long, with flat ends, 

 and that it strikes quite fair against an absolutely un- 

 yielding surface. The latter condition could not be fulfilled 

 in practice, because there is no substance more rigid than 

 steel. So far as the effects on the rod are concerned, how- 

 ever, it can be fulfilled by making two rods, moving with 

 equal velocities in opposite directions, collide end on ; and 

 this device has been used in the laboratory for imitating 

 the effect of impact against an unyielding surface. We 

 have to consider how long it takes to stop the rod under 

 such conditions. When the end first strikes it is pulled 

 up dead, just as in the case of the lead bullet, only it does 

 not now flow out sideways. The pressure, however, set up 

 at the end of the rod cannot be instantaneously transmitted 

 through it, and consequently the hind parts do not at once 

 feel this pressure, but continue to move on as before. The 

 transmission of the pressure takes place with the velocity 



NO. 2207, VOL. 88] 



of sound, which for steel is about 17,000 feet per second, 

 and it takes, accordingly, 1/20,000 part of a second before 

 the pressure has been transmitted throughout the 10 inches 

 length of the rod. .\ wave of pressure is initiated at 

 the first contact and travels along the rod. At any instant 

 the part of the rod which has already been traversed 

 by the wave will be at rest and in compression, while 

 the remainder which has not yet been reached by the 

 wave, and accordingly as yet knows nothing of the 

 impact, will still be moving forward with the old 

 velocity. Each section continues to move on until the 

 wave reaches it, when it is stopped with a jerk, the 

 sections thus pulling up successively until the whole rod is 

 at rest, which happens when the wave has travelled to the 

 free end. From the momentum of the rod, and the time 

 taken to stop it, the pressure can be calculated by the use 

 of the principles already illustrated. Thus a rod 10 inches 

 long is stopped, as we have seen, in 1/20,000 second, and 

 if it be moving with the moderate velocity of 20 feet per 

 second, the pressure required to pull it up in this time is 

 15 tons per square inch. This pressure is constant 

 throughout the impact, and it is obvious that here again 

 the intensity of pressure is dependent only upon the 

 velocity, and not on the weight of the rod ; for if with 

 the same velocity the length is increased, the correspond- 

 ing increase of momentum to be destroyed is cancelled by 

 the greater time required for the transmission of the 

 pressure wave, and if the area is increased the total 

 pressure is merely increased in proportion, the pressure 

 per unit area remaining the same. For a hard elastic body 

 the pressure is proportional to the velocity, a principle 

 which is probably generally applicable in the initial stage 

 of all impacts. 



At the instant of greatest compression, when the rod is 

 reduced to rest, it is like a compressed spring, and there 

 being no pressure acting at its free end to keep it com- 

 pressed, it proceeds to expand again. Starting at the free 

 end, a wave of expansion travels down the rod, the several 

 portions being successively jerked into motion with 

 approximately the original velocity. The whole process of 

 restoring motion to the rod is completed when this wave 

 reaches the impinging end, when the rod rebounds as a 

 whole with the original velocity. The whole time of con- 

 tact is, then, that taken by a wave of sound to travel twice 

 the length of the rod. Here, again, by electrical measure- 

 ment of the time of contact, it is possible to check the 

 theory. It is found that the actual time is longer than 

 that predicted. This is due to the fact that one cannot in 

 practice make the rods hit absolutely true all over the 

 ends ; they strike at one point first, and the metal near 

 that point has to be flattened out before the ends come 

 into contact all over and initiate the simple plane pressure 

 wave of the theory. The complete analysis of the dis- 

 crepancies between theory and experiment so caused was 

 long a puzzle to physicists interested in these matters. It 

 was finally effected' by Mr. J. E. Sears, who determined 

 mathematically the corrections necessary on this account, 

 and submitted his theory to experimental test with entirely 

 satisfactory results. . 



Another simple instance of the propagation of waves 

 along rods illustrates a point of importance in regard t. 

 the general effect of blows. Instead of maintaining th 

 pressure during the whole passage of the wave up and 

 down, as in the end-on impact, a pressure is suddenly 

 applied to one end, maintained for a short time, and then 

 removed. A corresponding pressure wave traveU along 

 the rod. Each portion of the rod is only stressed or in 

 motion during the passage of the wave over it. and .ifter 

 the passage of the wave it is left with a certain forward 

 displacement, but withoufany velocity or stress. F"'-'''^- 

 more the whole momentum of the blow is concentrated in 

 the short length of the rod covered bv the wave. On it» 

 arrival at the other end the wave is reflected, but the 

 Reflected wave is a wave of tension. As it comes back 

 the head of the tension wave is at first wholly or P«J 'a"y 

 neutralised bv the tall of the pressure wave, but after a 

 SI clears this, and the rod is then put into tension 

 of amount equal to the original pressure. I there be a 

 crack or weSk place in the rod at a sufficient distance 

 from the free end. the pressure wave will pass over it 

 prTtirn Iv unrhanged ; but on the arrival of the reflected 



