REFLECTION OF DIVERGING WAVES 375 



and 



A7£' — A(7 = — {w — (/)cA< — i^t. 



dx pc dt 



From which 



Aw = —c-^At — At, 



dx pc dt 



A^ = —c — At. 

 dx 



Hence the velocity is the same as when — is zero but iv is changed by 



— — — A/. But the only way in which w can change with q constant is 

 pc at 



by adding waves of equal amplitude propagating in opposite directions, so 



that their contributions to w are equal and those to q are equal and opposite. 



T f)f 



From (10) this involves an increment of — of — 2 — A/ or a time rate of change 



2 dt 



of —2 — .This agrees with (6), from which it is evident that the presence of 

 dt 



— alters — from what it would otherwise be by — -. But, since q is 



dt dx ^ pc^dt ' ^ 



unchanged, the velocities at .v -| — ~ and x — — - are increased by — — ^—Ax 



2 2 pc- dt 



1 /)/ 

 and — - — A.v. The first is the velocity associated with an auxiliary wave which 

 pc- dt ^ ^ 



propagates in the positive direction of x, and the second that of one which 



propagates in the negative direction, that is a reflected wave. Hence the 



1 ri{ 



medium generates a reflected wave of — - ^ per unit length in the direction 



pc- dt 



of propagation. 



The Reflection of a Progressive Diverging Wave 



So far attention has been confined to a single point. If a continuous dis- 

 turbance is being propagated, it is important to know how the waves reflected 

 at different points combine, for it is conceivable that they may interfere 

 destructively. From the standpoint of the application to be made of these 

 results in a companion paper, the case of most interest is that in which energy 

 is propagated outward from a central generator as a sinusoidal wave of 

 finite amplitude, beginning at time zero. Near the center, the wave of dis- 

 placement will include radial as well as tangential components. As the radius 



