Lord Rayleigh's Acoustical Observations, 279 



and may be regarded as a vibration of frequency J (^1 + ^2)? 

 and of amplitude 2 cos ir (% —n 2 )t. Hence, in passing through 

 zero, the amplitude changes sign, which is equivalent to a 

 change of phase of 180°, if the amplitude be regarded as al- 

 ways positive. This change of phase is readily detected by 

 measurement in drawings traced by machines for compound- 

 ing observations. If a force of the above character act upon 

 a system whose natural frequency is ^(n 1 + n 2 ), the effect 

 produced is comparatively small. If the system start from 

 rest, the successive impulses cooperate at first, but after a time 

 the later impulses begin to destroy the effect of former ones. 

 The greatest response is given to forces of frequency % and 

 n 2 , and not to a force of frequency ^(^1 + ^2)- 



On the other hand, when a single vibration is rendered in- 

 termittent by the periodic interposition of an obstacle, there 

 is no such change of phase in consecutive revivals. If a force 

 of this character act upon an isochronous system, the effect 

 is indeed less than if there were no intermittence ; but as all 

 the impulses operate in the same sense without any antago- 

 nism, the response is powerful. An intermittent vibration 

 or force may be represented by 



2(1 -f cos 2irmt) cos 2Trnt, 

 in which n is the frequency of the vibration, and m the fre- 

 quency of intermittence. The amplitude is always positive, 

 and varies between the values and 4. By ordinary trigo- 

 nometrical transformation the above expression may be put 

 into the form 



2 cos 27rw£ + cos 27r(n + m)t + cos 2ir(n—m)t ; 

 which shows that the intermittent vibration is equivalent to 

 three simple vibrations of frequencies n, n + m, and n—m. 

 This is the explanation of the secondary sounds observed by 

 Mayer. When m is equal to J n, n + m:n—m — 5 : 3, or the 

 interval between the secondary sounds is a major sixth. The 

 frequency of the resultant sound is m (that is, \n) : and its 

 pitch is two octaves below that of the original vibration. 



In the Edinburgh Proceedings for June 1878, an experi- 

 ment similar to Mayer's is described by Professors Crum 

 Brown and Tait, and is explained in the above manner. 



If the intensity of the intermittent sound rise more sud- 

 denly to its maximum, we may take 4 cos 4 irmt cos 2irnt ; and 

 this may be transformed into 



I cos 2irnt + cos 2ir(n + m)t + cos 27r(n—m)t 

 + \ cos 2ir(n + 2m)t + \ cos 2ir(n—2m)t. 

 There are now four secondary sounds, the frequencies of 



