CHAP, iv.] HEARING. 1005 



ahead of those belonging to the other; hence a time will come 

 when, while the impulse of one sound is tending to drive a 

 particle in one direction, say forwards, the impulse of the other 

 sound is tending to drive the same particle in the other direc- 

 tion, backwards. The result is that the particle will not move, 

 or will not move so much as if it were subject to one impulse 

 only, still less to both impulses acting in the same direction; 

 the vibrations of the particle will be stopped or lessened, and 

 the sensation of sound to which its vibrations are giving rise 

 will be wanting or diminished $ the one sound has more or less 

 completely neutralized or "interfered" with the other, the crest 

 of the wave of one sound has more or less coincided with the 

 trough of the wave of the other sound. Conversely at another 

 time, the two impulses will be acting in the same direction on 

 the same particle, the movements of the particle will be inten- 

 sified, and the sound will be augmented. And the one condition 

 will pass gradually into the other. The repetitions of increased 

 intensity thus brought about are spoken of as beats. 



The length of the interval at which the beats recur will 

 depend on the difference of period of the two sounds in relation 

 to the actual period or pitch of each. It may be stated gener- 

 ally that the number of beats in a second is equal to the differ- 

 ence between the number of vibrations per second of the two 

 sounds; thus two very low pitched tuning-forks, vibrating 

 respectively at 64 and 72 a second, will give 8 beats a second, 

 and two very high pitched tuning-forks, vibrating respectively 

 at 4224 and 4752 a second, will give 528 beats a second; but in 

 this respect there are complications which we cannot consider 

 here. , 



Beats are produced when the periods of the coincident sounds 

 are not exact multiples of each other. When the periods are 

 exact multiples no beats occur; two tuning-forks, for instance, 

 the period of one of which is exactly double that of the other, 

 give rise to no beats when sounding together; and so in other 

 instances. 



By beats then a continuous musical sound may be broken 

 up into a series of discontinuous sounds. When the beats are 

 repeated a few times only in a second the discontinuous sounds 

 give rise to discontinuous sensations; we hear the separate beats. 

 But if the beats are repeated sufficiently rapidly the successive 

 sensations are fused in one, we cease to hear the beats as such, 

 though we have other evidence that the beats continue to be 

 produced. Just as a series of simple vibrations when repeated 

 sufficiently rapidly, say 40 times a second, gives rise, by summa- 

 tion, to a single musical sound, to a tone, so a series of groups of 

 vibrations, each group corresponding to the interval between two 

 beats, gives rise when the groups follow each other rapidly, by 

 a similar summation, to a continuous sensation. And, though 



