84 THE STUDY OF SPEECH CURVES. 



moment at which y begins its positive values (indicated by j/= + 0), then 

 q = 0, and the equation becomes 



y = a.sin-nii' (2) 



If the distance from the origin of coordinates to y=-hO be denoted 

 by r, then 



r = |r. (3) 



A sine curve with the phase q= — ^ is the same as a cosine curve 

 with the phase 0, or 



If the " frequency, " or number of vibrations per second, is n= IjT, then 



y = a.8m27:nt. (4) 



If the number of vibrations in 2;r seconds is k, then 



y = a. sin kt. (6) 



According to Fourier's theorem any periodic function of t may be 

 expressed by a constant, plus the sum of a harmonic series of sines and 

 cosines where the amplitudes receive the necessary values and the series 

 is sufficiently extended. Thus, 



+ tti. COS — t + (h. COS jj,t-\-a3.co8—^t+ . . . 



(7) 

 + b^.sin —t + h.sin ^t + h.sm—t+ . . . 



Here C is the ordinate for the mean value of the function (height 

 of the curve axis above the X-axis), T is the period of the function to 

 be expressed, T, \T, IT, . . . are the harmonic series of periods, and 

 fli, tti, tts ,. . . &i, 62, &3, . . . are the ampUtudes of the sinusoids belonging 

 to the series. 



