PHOBLEM OF THE SOUNDS OF PIPES. 30 



sisted in vibrations of some kind ; but to determine the nature and laws 

 of these vibrations, and to reconcile them with mechanical principles, 

 was far from easy. The leading facts which had been noticed were, 



J O 



that the note of a pipe was proportional to its length, and that a flute 

 and similar instruments might be made to produce some of the acute 

 harmonics, as well as the genuine note. It had further been noticed, 1 

 that pipes closed at the end, instead of giving the series of harmonics i, 

 i, i, i, <fec.., would give only those notes which answer to the odd 

 numbers i, i, , &c. In this problem also, Newton 2 made the first 

 step to the solution. At the end of the propositions respecting the 

 velocity of sound, of which we have spoken, he noticed that it appeared 

 by taking Mersenne's or Sauveur's determination of the number of 

 vibrations corresponding to a given note, that the pulse of air runs 

 over twice the length of the pipe in the time of each vibration. He 

 does not follow out this observation, but it obviously points to the 

 theory, that the sound of a pipe consists of pulses which travel back 

 and forwards along its length, and are kept in motion by the breath of 

 the player. This supposition would account for the observed depend- 

 ence of the note. on the length of the pipe. The subject does not 

 appear to have been again taken up in a theoretical way till about 

 1760; when Lagrange in the second volume of the Turin Memoirs, 

 and D. Bernoulli in the Memoirs of the French Academy for 1762, 

 published important essays, in which some of the leading facts were 

 satisfactorily explained, and which may therefore be considered as the 

 principal solutions of the problem. 



In these solutions there was necessarily something hypothetical. In 

 the case of vibrating strings, as we have seen, the Form of the vibrating 

 curve was guessed at only, but the existence and position of the Nodes 

 could be rendered visible to the eye. In the vibrations of air, we can- 

 not see either the places of nodes, or the mode of vibration ; but several 

 of the results are independent of these circumstances. Thus both of 

 the solutions explain the fact, that a tube closed at one end is in unison 

 with an open tube of double the length ; and, by supposing nodes to 

 occur, they account for the existence of the odd series of harmonics 

 alone, i, 3, 5, in closed tubes, while the whole series, i, 2, 3, 4, 5, <fcc., 

 occurs in open ones. Both view r s of the nature of the vibration appear 

 to be nearly the same ; though Lagrange's is expressed with an analy- 

 tical generality which renders it obscure, and Bernoulli has perhaps 



D. Bernoulli, Berlin. Mem. 1753, p. 150. 2 Prlncip. Schol. Prop. 50. 



