76 Mr. V. Lough on the Beating 



results canuot be relied on quantitatively, but they show 

 clearly enough the cyclical character of the variation and 

 the changing proportions of the partial tones with alteration 

 of pressure. The results of the harmonic analysis from 

 measurements of selected waves on each of these curves 

 are exhibited for comparison with the results of the corre- 

 sponding ear-tests in the accompanying table. 



It will be seen from the figures in this table that instead 

 of increasing steadily and then falling abruptly as might be 

 expected, the amplitude of the fundamental after rising to a 

 maximum falls off gradually with increasing wind-pressure. 

 If beats occur at all in this partial, which is doubtful, they 

 occur only over a short range. The octave increases 

 gradually and develops well-marked beats before settling 

 down to the final steady tone, while the twelfth increases to 

 a maximum and then falls off and disappears with the 

 fundamental ; the beats are most prominent in this partial. 



3. Discussion of the Results. 



Following on the analogy pointed out by Helmholtz 

 between the form of the vibration of the air in narrow 

 flue pipes and that of a violin string under the action of the 

 bow, and in view of the comparison already suggested 

 between the beating tones of overblown pipes and the similar 

 effects obtained with howed stringed instruments in certain 

 cases, we should be led to seek an explanation of the beating- 

 tones in the departure of the free periods of vibration of the 

 air-column from harmonic relation and a failure of the air-jet 

 to supply sufficient energy for the continuous maintenance of 

 all the possible modes of vibration. There are, however, 

 certain difficulties in accepting this view in its entirety. It 

 is true that the end corrections being not quite independent 

 of wave-length have the effect of rendering the partials 

 slightly inharmonic, and a similar effect may also arise from 

 the elasticity of the walls of the pipe. But beyond this 

 point the analogy does not seem to help us, for it is difficult 

 to suppose that the air-jet which at normal pressure exhausts 

 only a small fraction oE its energy in sustaining the full tone 

 of the pipe, can become less effective at higher pressures. 



Again, though the records of the actual vibrations show a 

 cyclical variation, they do not at any stage show the ,very 

 strongly marked fluctuation amounting almost to periodic 

 disappearance of the fundamental vibration which is cha- 

 racteristic of the records with bowed instruments obtained 

 at the wolf-note pitch. 



We are therefore forced to seek an explanation of the 



