28 BELL SYSTEM TECHNICAL JOURNAL 



waves spread out from a plane source in an infinite homogeneous space. 

 However, the analogy is not complete. The inward bound spherical 

 wave cannot exist without an appropriate receiver of energy at the 

 origin. In absence of such a receiver, the energy condenses at the 

 origin and spreads outward again. The result of interference between 

 two such progressive waves will be called the "internal" spherical 

 wave.^ It is natural to regard a thin spherical source in an infinite 

 homogeneous medium as an analogue of a thin plane source and to 

 consider the waves on the two sides of such a spherical source as the 

 mates. In accordance with this idea the ( + ) and the ( — ) signs are 

 used to distinguish between the waves produced by a source on its 

 two sides rather than to indicate "progressive" waves moving in oppo- 

 site directions. This attitude is not only a possible and a natural 

 attitude but almost a necessary one in view of the fact that no gen- 

 erally applicable criterion is known by which "progressive" waves 

 could be identified in any particular case. As often happens, in simple 

 situations there is no need for arguing as to which attitude is the more 

 proper one; thus the waves on the two sides of a plane source in an 

 infinite homogeneous medium are two progressive waves moving in 

 opposite directions. 



The field of the internal spherical wave is 



^ iwuA I . , cosh ar , sinh ar\ . „ 



Eg- = -^ — smh (jr ^-^— sm 0, 



Iwr \ ar a-r- ) 



^ TfA / sinh ar u \ a 



Er~ = — r, cosh ar cos d, 



■wr- \ ar J 



^^ a A ( sinh ar u \ • a «% 



H^- = 7. — cosh ar sm d. ^ 



27rr \ ar I 



The corresponding impedance is then 



. , cosh ar . sinh ar 



„ smh ar ir-^ — 



EfT or a-r^ 



H^~ , sinh ar 



cosh ar ■ 



ar 



Close to the origin we have approximately 



2 



z,- = 



{g + zcoe)r' 



* If the medium is non-dissipative, this wave is a standing wave; but, in general, 

 it is simply a combination of two progressive waves in such proportions that the field 

 is finite at the origin. 



