Propagation of Disturbances in Elastic Media. 465 



part of the medium considered and the time for which it acts. 

 The force will involve the particular elasticity of the medium 

 concerned in the disturbance ; and the time will be inversely 

 proportional to the velocity of propagation. 



By equating the momentum generated to the product of 

 the force and the time for which it acts, we get an equation 

 by which to find the velocity of propagation. 



In the case of the propagation of a series of waves, they 

 may be considered as a succession of single disturbances, and 

 the velocity of propagation will be the same as for a single 

 disturbance. 



Longitudinal Disturbance, 



Suppose a long cylindrical uniform bar of cross section S. 

 Let a compression be created in this bar at one end, and pro- 

 pagated along it from left to right. We may imagine the 

 passage of the compression to take place by supposing each 

 portion of the bar, as soon as the compression reaches it, to 

 press forward on the next adjoining portion and compress it, 

 returning itself to its natural condition, the whole of the bar 

 to the left of the compression being left in its natural condition. 



Let E be the Young's modulus of the bar, d its density. 

 Suppose a length x to be involved at each instant in the com- 

 pression, and let I be the amount of the compression, so 

 that each point of the bar travels forward by a distance I as 

 the compression passes through it. Let V be the velocity 

 with which the compression travels. 



Consider two planes A and B drawn at right angles to the 

 bar and fixed in space. Then while the compression is passing 

 across A, the mean pressure at A in excess of that at B is 



E - . And the time taken for the compression to pass A 

 x 



. x 

 ls y- 



Thus the momentum generated in the space between A and 

 Bis 



ES -rv 



But in each second a length V of the bar is displaced for- 

 ward by a distance I. Thus we have for the momentum 

 Phil. Mag. S. 5. Vol. 31. No. 193. June 1891. 2 L 



