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LVII. On the Partial Tones of Boiced Stringed Instruments* 

 By C. V. Raman, M.A., Palit Professor of Physics in the 



Calcutta University*. 



1. Introduction. 



ONE of the outstanding questions in the acoustics of the 

 violin family of instruments which has not us yet been 

 fully cleared up, is the manner in which the tones elicited by 

 bowing depend on the position of the bowed " point " on the 

 string. The problem is to some extent complicated by the 

 existence of other variable factors influencing the character 

 of the vibrations excited, e. g. the bowing pressure and speed, 

 and the width of the region of contact between the bow and 

 string. In a monograph of which the first part has been 

 recently publishedt, I have attempted a systematic treatment 

 of the mechanical theory of bowed stringed instruments, and 

 have dealt with various problems relating to it. It is pro- 

 posed in the present paper to apply some of the results 

 contained in the monograph in order to discuss the variation 

 of the amplitudes and phases of the partial vibrations with 

 the position of the bowed point within the musical range of 

 bowing. 



We may here assume that in the cases of musical interest, 

 the steady vibrations of the string excited by the bow r have 

 approximately the character of the simple Helmholtzian type 

 in which the vibration-curve of every point on the string is a 

 perfect two-step zig-zag; and the problem is to rind the 

 nature and extent of the small deviations from this form of 

 vibration depending on the position of the bowed point and 

 other variable factors. If the vibrations were always exactly 

 of the simplest Helmholtzian type irrespective of the position 

 of the bowed point, it can readily be shown that for a given 

 speed of the bow, the amplitudes of all the components of the 

 vibration would be inversely proportional to the distance of 

 the bowed point from the bridge, and would thus increase 

 indefinitely according to a hyperbolic law as the bow is 

 brought nearer the bridge ; the ratios of the amplitudes, and 

 the relative phases of the partials, would be independent 

 of the position of the bowed point. In practice, however, 

 we know that the foreooino- statement does not correctly 

 represent the facts. For, when the bow is applied at a point 

 of aliquot division of the string, e. g.-dt a point distant 1/5 

 or 1/6 or J/7 &c. of the length of the string from the 



* Communicated by the Author. 



t Bulletin No. 15 of the Indian Association for the Cultivation of 

 Science, Calcutta, 1918, pages 1-158. 



