462 BELL SYSTEM TECHNICAL JOURNAL 



Now, a limited wave-train is equivalent to an infinity of infinitely 

 long wave-trains, with an infinite variety of frequencies. The ampli- 

 tudes and the frequencies of these constituent waves are chosen so 

 that the waves reinforce one another over the length of the signal, 

 counteract one another for all time before and after the signal. To 

 choose them thus is always mathematically feasible, whatever shape 

 of signal be prescribed. Whether these constituent waves should be 

 regarded as "physically real" is a question that was discussed long 

 before the days of quantum theory and other modern puzzles. Any- 

 how, by taking them as such, one arrives at verifiable statements 

 about the signals, and this is all that matters. 



Let us now conceive the signal as a chopped-off segment of a sine- 

 wave-train of frequency /o; and let us compare its travel with the 

 travel of a limitless wave-train of frequency / put equal to /o. One 

 feels that the signal ought to follow the same path through the iono- 

 sphere as would the limitless train, and ought to move along that 

 path with the same speed as would the wave-crests of the limitless 

 train. This is true if, but only if, the speed u of the wave-crests in 

 unlimited trains is independent of /. But when wireless waves are 

 traveling through the ionosphere, u varies very much with/, according 

 to a law which will later be worked out; and this makes a remarkable 

 difference. 



The difference as to path is not serious. The infinite wave-trains 

 which form the signal are most intense at frequencies very close to/o, 

 and this is sufficient to make the signal follow nearly (though not 

 without some deviation and distortion) the path which the infinite train 

 of frequency /o would follow by itself. We may therefore regard the 

 broken lines of Fig. 1 as the paths of signals or of waves, indifferently. 



The difference as to speed is serious. It is not reduced by the 

 preponderance of component wave-trains very close to /o, and it does 

 not tend to vanish as this preponderance is increased by lengthening 

 the signal. It remains serious in the artificially-simplified case of just 

 two component wave-trains of small frequency-difference A/, where 

 the signals become the "beats" well known in acoustics and in radio. 

 In this case the beat-speed or signal-speed approaches a limit as A/ 

 approaches zero. This limit, the "group-speed" denoted by v, is 

 always used for the signal-speed, though for actual signals it is but 

 an approximation, and the signals themselves become distorted as 

 they travel. Contrasted with v is the "wave-speed" or the "phase- 

 speed" of the wave-crests of the unlimited wave-trains, already 

 denoted by u. 



