466 BELL SYSTEM TECHNICAL JOURNAL 



amounts to saying that the phase-speed is infinite when N = Nc, 

 imaginary when N > Nc. However, the concept of phase-speed is 

 of such a quaHty of abstractness, that even these statements imply 

 nothing absurd in the physical situation. The signal-speed itself 

 remains safely finite and real. 



The signal-speed is strictly indefinite, since the signal distorts itself 

 as it proceeds. However, the practice is to identify it with the group- 

 speed V, which, as I intimated (page 462), is the speed of the beats 

 formed by two superposed wave-trains difTering infinitesimally in 

 wave-length, each such beat being a very special type of signal. 

 The formula is, 



. = „-XW<iX) = «/(l-^^)- (6) 



It is difficult to visualize or derive without a diagram,* but the deriva- 

 tion may be summarized as follows. Imagine two superposed wave- 

 trains of phase-speeds u and u + du, wave-lengths X and \ -\- d\; 

 consider two consecutive wave-crests A, A' of one and two consecutive 

 wave-crests B,B' of the other; transpose temporarily to a frame of 

 reference in which the former wave-train is stationary. At a certain 

 place and time A and A' will coincide, and the maximum of one of 

 the beats will be right there. Let the time dXjdu elapse; when it 

 has elapsed, the crests B and B' will be coinciding and the maximum 

 of the beat will have moved on by one entire wave-length. The beat 

 therefore travels with speed Xdu/dX in the temporary and with speed 

 u — \{du/dX) in the original frame of reference (the minus sign is 

 evident when the reasoning is gone through in detail). 

 Combining (6) with (2) one finds: 



V — c~Ju; (7) 



the greater the phase-speed, the slower the signal ! Relativists will 

 be pleased to observe that according to this formula, the signal never 

 attains any speed greater than c; students of quantum mechanics 

 may be misled by its superficial resemblance to a formula relating 

 phase-speed to group-speed for de Broglie waves, with which it has 

 nothing to do. Students of the ionosphere should remember its 

 approximative character. Almost all that needs to be known for the 

 purposes of this article is, that as a signal climbs into the ionosphere 

 it goes more and more slowly, the nearer N approaches to that value 

 Nc where the signal finds its ceiling. 



^ Cf. this journal, 9, 173 (1930), or my Introduction to Contemporary Physics, 

 2nd edition, p. 147. 



