376 BELL SYSTEM TECHNICAL JOURNAL 



increases the radial components become relatively negligible. We shall 

 confine our attention to this outer region, where, in the absence of reflection, 

 the propagation differs from that of a plane wave only in that the amplitude 

 varies inversely as the radius. We shall neglect the effect of any reflections 

 on the outgoing wave, and calculate the resultant reflected wave at a radius 

 Ti as a function of the time and so of the radial distance r the wave front 

 has traveled. 



If the outgoing wave were of infinitesimal amplitude, its velocity q^ 

 could be represented by 



Oo = - <3o sin {icl — kr), (11) 



r 



for values of r < ct, and by zero for r > ct, where Qo is the amplitude at 

 some reference radius ro . The sine function is chosen to avoid the necessity 

 of an infinite acceleration at the wave front, as would be required by a 

 cosine function. When the amplitude is finite this wave suffers distortion 



due to the fact that k which is equal to - varies slightly with the variations 



in the instantaneous value of c. However, these will be small and, since 

 fluctuations in velocity alone do not cause reflection, we shall neglect them. 

 The procedure is to make use of ^o to calculate the reflected wave incre- 

 ment generated in a length Ar' at a radius r', calculate the amplitude and 

 phase of this at a fixed point r^ <r', and at ri integrate the waves received 

 there for values of r' from ri to the farthest point from which reflected waves 

 can reach ri at the time t under consideration. 



To find the reflected wave generated in a length Ar' at r', we have from 

 above that its velocity 



Aq' = l^^Ar'. 



From (21a), (19a) and (l7a) 



1 Fi 



2 — 



pc 



(1— a I (p dt 1 



where tjo and a are constants of the medium given by (7a) and (15a). From 

 (18a) 



di 

 80 



= — arjo^ / <pdt, 



