174 BELL SYSTEM TECHNICAL JOURNAL 



Now if the two component waves advance with equal speed, the 

 beats are simply carried along with a speed equal to theirs. But if the 

 velocities of the two component waves are not the same, then the 

 velocity of the beats is not the same as either, nor the mean thereof. 

 It is in fact something totally different. 



To see this, imagine that you are moving along with one of the sine- 

 waves; for definiteness, that you are riding on the crest B of the train 

 with the shorter waves (Fig. 3). At a certain moment, say / = 0, 

 it coincides with a crest A of the other sine-wave, and you are at the 

 top of the beat. Meanwhile the other train is moving relatively to 

 the first; for definiteness suppose that the longer waves move faster, 

 so that relatively to the shorter they are gliding upward. After a cer- 

 tain time they have gained on the shorter waves by a distance AX, the 

 difference between the two wave-lengths. But when this time has 

 elapsed, the top of the beat is no longer where you are, but where the 

 crest B' of the first train coincides with the crest A' of the second. 

 It has dropped back through the distance X, while the second wave 

 train was getting ahead by the distance AX. Perhaps it will be easier 

 to realize that while the second wave train is gaining on the first by X, 

 the beat is dropping back by the distance D between consecutive beats; 

 by equation (17) this comes to the same thing. 



Therefore when the longer waves travel faster than the shorter, the 

 beats travel more slowly than either. If the longer waves were the 

 slower, the beats would travel more rapidly; but this case is never 

 realized in nature, not at least with light-waves '' and waves of elec- 

 tricity^ and matter. 



We now deduce the formula for the actual value of the speed of the 

 beats. Denote by v and v -\- f^v the speeds of the two sine-waves of 

 which the wave-lengths are X and A -f- AX, respectively; by g the 

 speed of the beats. It is sufficient to put into notation what has just 

 been said in words. Relatively to the former wave train, the velocity 

 of the latter wave train is Av, that of the beats is {g — v). Relatively 

 to the former wave train, the latter moves a distance AX while the 

 beats are moving a distance X in the opposite sense, therefore with a 

 minus sign. Hence: 



{g - v)/M' = -X/AX (18) 



" The exception to this statement — the case of light having wave-lengths lying 

 within a region of anomalous dispersion of t he transmitting substance — has been an- 

 alyzed by Sommerfeld and L. Brillouin {A7in. d. Phys 44, pp. 177-202,203-240; 1^14) 

 who find that in this case the group-speed defined by (20) loses its physical im- 

 portance, and a segment of a wave train is transmitted with a speed never exceed- 

 ing the speed of light in vacuum. This appears to be related to the absorption 

 which always gees with anomalous dispersion. 



