Propagation of Disturbances in Elastic Media. 467 



Thus the successive layers still move towards the right, now 

 starting from the free end. Thus the compression is reflected 

 as a rarefaction. In the same way a rarefaction is reflected 

 as a compression. This is the case of reflexion without change 

 of sign. 



When a compression reaches a fixed end, the momentum 

 of the bar, from left to right, is more than destroyed at the 

 end, and a compression is reflected. In the same way a 

 rarefaction is reflected as a rarefaction. This is the case of 

 reflexion with change of sign. 



Case of Two Media. — Suppose we have two bars of cross- 

 section S joined end to end, being the common surface. 



Let E x , di, Vi ; E 2 , d 2 , Y 2 denote the Young's modulus, den- 

 sity, and velocity of wave propagation in the two media. 

 Suppose a compression producing a displacement I to the 

 right to be propagated from left to right in A C. Let this 

 produce a reflected disturbance of A C, causing displacement 

 V to the right. Then, since the bars on both sides of C 

 return to their natural conditions after the passage of the 

 disturbance, the disturbance transmitted to C B causes dis- 

 placement l + V to the right. Now the momentum in the 

 space A B is the same just before and just after the disturb- 

 ance in A C reaches C. Thus we have 



S^Vi = $1%% + S {I + V) d 2 Y 2 ; 



... (i-qd.Y^ii+i^d.v,. 



Thus V will have the same sign as Z, or there will be 

 reflexion without change of sign, if d^Sf l is > d 2 Y 2 , or c^Ej 

 is >d 2 ~E 2 . 



V will have the opposite sign to I, or there will be reflexion 

 with change of sign, if d 1 Y ] is < d 2 Y 2 , or d l E 1 is < d 2 T$ 2 . 



i i Transverse Disturbance. 



Suppose a uniform bar of cross-section S ; and let M be its 

 rigidity, and d its density. Let a transverse disturbance be 

 sent along it from left to right. Let the disturbance be such 

 that each point of the bar is displaced upwards by a distance 

 ■I as the disturbance passes it. Let $ be the length involved 



2 L 2 



