438 MR. B. HOPKINSON ON MEASURING THE PRESSURE PRODUCED IN THE 



by the successive ordinates of the curve as they pass over it. Thus the curve also 

 represents the relation between the pressure at any point of the rod and the time, 

 the scale being such that one inch represents the time taken by the wave to travel 

 that distance which is very nearly ., 00 * 000 of a second. In particular the curve 

 giving the distribution of pressure in the rod along its length is, assuming perfect 

 elasticity, the same as the curve connecting the pressure applied at the end and the 

 time, the scale of time being that just given. 



The progress of the wave of stress along the rod is accompanied by corresponding 

 strain and therefore by movement. It is easy to show that the same curve which 

 represents the distribution of pressure at any moment also represents the distribution 

 of velocity in the rod, the scale being such that one ton per square inch of pressure 

 corresponds to about 1'3 feet per second of velocity. Until the wave reaches any 

 section of the rod that section is at rest. It is then, as the wave passes over it, 

 accelerated more or less rapidly to a maximum velocity, then retarded, and finally left 

 at rest with some forward displacement. In this manner the momentum given to the 

 rod by the application of pressure at its end is transferred by wave action along it, 

 the whole of such momentum being at any instant concentrated in a length of the rod 

 which corresponds, on the scale above stated (one inch = ., 00 ^ )OU second), to the time 

 taken to stop the bullet completely. Consider a portion of the rod to the right of any 

 section A (fig. l) which lies within the wave at the moment under consideration. 

 The pressure has been acting on this portion since the wave first reached it, that is 



OA 

 for a time represented by the length OA and equal to -y^- where V is the velocity of 



propagation. The momentum which has been communicated to the part under 

 consideration is equal to the time integral of the pressure which has acted across the 

 section A, that is to the shaded area of the curve in the figure. The portion of the 

 rod to the right of the section is continually gaining momentum at the expense of the 

 portion to the left while the wave is passing, the rate of transfer at any instant being 

 equal to the pressure. 



When the wave reaches the free end of the rod it is reflected as a wave of tension 

 which comes back with the same velocity as the pressure wave, and the state of stress 



in the rod subsequently is to be deter- 

 mined by adding the effects of the direct 

 and f the reflected waves. Now suppose 

 that the rod is divided at some section, B, 

 near the free end (fig. 2), the opposed 

 surfaces of the cut being in firm contact 



and carefully faced. The wave of pressure travels over the joint practically 

 unchanged and pressure continues to act between the faces until the reflected 

 tension wave arrives at the joint. The pressure is then reduced by the amount 

 of the tension due to the reflected wave and as soon as this overbalances at 



