466 Mr. W. T. A. Emtago on the Velocities of 



generated between A and B, while the compression is tra- 

 velling between A and B, the expression 



Vl8d. 

 Equating these expressions, we get 



~ d 



The same amount of momentum is generated between A 

 and B by the passage of the compression across B, thus leaving 

 the portion A B at rest. 



In the same way a rarefaction may be propagated by the 

 successive portions of the bar moving to the left by a dis- 

 tance I. The momentum generated in A B by the passage of 

 the rarefaction across A, and destroyed by its passage across 

 B, is in this case from right to left. 



It has been shown, so far, that the disturbance is propa- 

 gated with a mean velocity A/ -7. But we may show that 



each portion of it is propagated with the same velocity, or 

 that it travels unchanged in form, by applying the same con- 

 siderations to any portion of the disturbance instead of to the 

 whole of it. 



Consider a portion of the disturbance of length x to be 

 crossing the plane A with velocity V. Let I be the amount 

 of compression in this portion. I is positive or negative 

 according as the length x is compressed or extended. The 

 passage of this portion across A generates momentum in A B, 



I x 

 from left to right, equal to E S . — • ~. Also the quantity of 



x v 



momentum generated in A B by this passage, supposing the 

 part of the disturbance in front of x to travel unchanged in 

 velocity and form, is YlSd. 



Thus, if we suppose the entire disturbance, of any form, to 



/E 



be moving with uniform velocity \ / -y, the forces brought 



into play at each point are just the necessary forces to keep 

 up the propagation unchanged. With any other velocity of 

 the disturbance, or of any part of it, this would not be the case. 

 This is therefore the velocity of the disturbance, and it travels 

 unchanged in form. 



Reflexion of Disturbances. — When a compression, moving 

 from left to* right, reaches the free end of a bar, the bar 

 possesses momentum from left to right which is not destroyed. 



