118 



Mr. H. Knapman. [June 6, 



"An Experiment Illustrating Harmonic Undertones." By 

 HERBERT KNAPMAN, B.A., Fellow of Emmanuel College, 

 Cambridge, and Assistant Lecturer in Mathematics at 

 University College, Heading. Communicated by GEORGE J. 

 BURCH, D.Sc., F.RS. Keceived June 6 Eead June 16, 1904. 



An Experiment Illustrating Harmonic Undertones. 



The object of this note is to describe a phenomenon of which, as 

 far as I have been able to learn, no published account has hitherto 

 been given, although perhaps its extreme simplicity renders it unlikely 

 that it has altogether escaped detection. If a vibrating tuning-fork 

 is pressed against a light object, such as a piece of paper or a stretched 

 string, this object will in general follow the vibrations of the fork 

 until they die away, remaining in contact with the fork all the time. 

 This method of using a tuning-fork is mentioned in Lord Rayleigh's 

 "Theory of Sound," 133. If, however, a piece of paper is touched 

 lightly by the vibrating fork, the elastic force opposing displacement 

 of the paper may be so small that it does not remain in contact with 

 the fork, but is driven away by the tap which it receives. The paper 

 may return in time to be struck by the fork before the latter has 

 executed a complete vibration, and the process being repeated again 

 and again the paper will vibrate with the same period as the fork, but 

 with a motion which is not simple-harmonic, on account of the 

 irregularity caused by the blows. The result of this is that the paper 

 emits a note in which the harmonic overtones are of considerable 

 importance, and which, therefore, resembles the note of a bowed violin 

 string rather than the almost pure tone of the tuning-fork. This 

 resemblance is easily perceived, as is also the change to the approxi- 

 mately pure tone which takes place when contact ceases to be broken, 

 on account of increase of pressure or falling off in the amplitude of 

 vibration of the fork. (The vibration of the air in the neighbourhood 



the fork has also an effect on the motion of the paper, but this 

 effect is probably so small in comparison with that of the blows that it 

 may here be omitted.) 



f the paper does not come within striking range of the fork again 

 the latter has nearly had time to execute two complete vibra- 

 te note given by the paper will be an octave below that of the 

 the vibrations of the paper have now a period which is 

 that of the fork. Thus, supposing the pitch of the fork to be 

 e paper will give the note c'. If the fork strikes the paper at 

 every third vibration the paper will give /, a twelfth below the note 

 Similarly contact at every four, five, six, or seven vibra- 

 U give c, A b , F, or a note a little sharper than D, the ratio of 



