72 Mr. V. Lough on the Beating 



construction of the differential equations are assumed to 

 be small, become, after its integration, large. There is no 

 reason whatever, therefore, for regarding the results as an 

 approximation to the actual conditions. Even in the stable 

 case the method lacks an accurate basis. 1} 



However, these remarks seem to me to be unduly severe, 

 after having borne them in mind for the past twenty-four 

 years. After a careful examination of a^ large number of 

 special problems, I have never found the method lead to 

 erroneous results, except in these critical cases (such as 

 the top-problem settled by Klein) ; and it must be remem- 

 bered that (even in the problems of Statics) the critical 

 cases are inevitably associated with the examination of terms 

 of higher order than is usually necessary. 

 I am, Gentlemen, 



Your obedient servant, 



T. J. Pa. Bromwich. 



*St. John's College, Cambridge, 

 4 August, 1921. 



VI. On the Beating Tones of Overblown Organ Pipes. 

 By V. Lough, B'.Sc, A.R.C.Sc* 



[Plate L] 



1. Introduction. 



ONE of the most interesting problems in Acoustics, 

 on which much has been written without arriving 

 at a complete solution, is the mechanism of the excitation 

 of " flue " or " flute " pipes by blowing. According to 

 Helmholtz, this is a simple matter of the wafting from 

 side to side of the blade-shaped air- jet into and out of 

 the mouth of the pipe under the influence of the oscillating 

 motion of the air within the pipe itself. In his book on 

 'The Sensations of Tone ' he explains how the air-jet 

 maintains these oscillations ; be suggests that, having no 

 appreciable stiffness, the jet follows the inward motion of 

 the air inside the pipe, and so delivers a puff of air at or just 

 after the period of maximum condensation. From the dis- 

 continuous nature of these impulses he deduces that the 

 forced vibrations maintained by them should, in the case 

 of narrow pipes having free periods in nearly harmonic 

 relation, produce a tone rich in upper partials, similar to 

 that of stringed instruments : a deduction which is fully 

 supported by experience. 



* Communicated by Prof. C. V. Raman, M.A. 



