WA VE- FORMS OP VO WEL TONES. 1221 



6. Such a compound wave is capable of being analysed into a number 

 of simple pendular vibrations, and each pendular vibration corresponds 

 to a simple tone, having a pitch determined by the periodic time of 

 the corresponding motion of the air (Ohm's law). 



7. It is evident that such combinations of simple waves may give 

 rise to an infinite variety of wave-forms, but, according to Fourier's 

 theorem, 1 " any given regular periodic form of vibration can always be 

 produced by the addition of simple vibrations, having vibrational 

 numbers, which are once, twice, thrice, four times, etc., as great as the 

 vibrational number of the given motion." 



8. If we know the amplitudes of the simple vibrations and the* 

 difference of phase, then any regularly periodic motion can be shown to 

 be the sum of a certain number of pendular vibrations ; in other words, 

 the compound wave may be analysed into its constituents. 



The Fourierian analysis of wave-forms. The method of applying 

 the Fourierian analysis may be shortly stated. 2 Suppose a single wave 

 (A A or BB) is taken and referred to rectangular axes (OX, OY), 

 abscissae of equal length are measured off along OX, ordinates are drawn 

 to meet the curves, and these are very accurately measured. For a 

 simple curve, such as AA, few ordinates suffice ; but if the curve is 

 complex in form, the number of measurements must be increased 

 until no considerable curvature 

 exists between the summits of 

 any two successive co-ordinates. 

 When a sufficient number of 

 ordinates has been taken, an 

 analysis carried to one-half that 

 number of partials, or even some- 

 what less, suffices to define the 

 curve. The mathematical process 

 for evaluating the amplitude and 

 phase of a certain number of par- 

 tials, say the first twelve, is given 

 by Fleeming Jenkin and Ewing * ^^^^ st ste " in 

 in their well-known paper. 3 A 



curve may be rebuilt graphically from its analysis, but the process 

 is laborious when the partials are numerous. If we know the pitch of 

 the fundamental tone, then the pitch of any partial is ascertained 

 by multiplying its vibrational number by the figure representing 

 the ratio of its amplitude to that of the fundamental tone. For 

 example, take the analysis of a Swedish i (resembling the French), 

 sung by Pipping at a pitch of 293 vibrations per second. 



An inspection of this figure shows that two reinforcements are very 

 pronounced, the one on the seventh and eighth, and the other on the 

 fourth partial. Each of these partials has to do with the resonance 

 cavity to which it corresponds, and in which it has been developed to 

 a greater or less extent. In the case of the vowel i, one of the reson- 



1 Donkin, "Acoustics," Fourier's theorem proved, pp. 65-71 ; illustrated, pp. 56-65. 

 See also Everett, " Vibratory Motion and Sound." 



2 The authors are much indebted to Dr. R. J. Lloyd for aid in the discussion of this 

 difficult subject. 



3 Trans. Roy. Soc. Edin., vol. xxviii. p. 750. 



