33 2 WAVE ANALYSIS. [XP, 



where* 



7? R 



^ = tan- 1 --^ ; <j> 3 = tan" 1 -^- ; etc. ( 2 b) 



XX ]_ ,/! o 



The first term in (2) represents the fundamental; the remain- 

 ing terms represent the harmonics of 3, 5, 7, etc., times the 

 fundamental frequency. Their amplitudes are given by C 19 C s , C 5 , 

 etc., and their relative phase positions by <f> lf <j> 39 <f>-, etc. The abso- 

 lute values of <f> 19 <j> 3 , <f> 5 , etc. (and the corresponding values of 

 A v AAi, etc., and B if B S) B 5f etc., but not of C 19 C 99 C S9 etc.), 

 depend upon the origin or point of reference from which angles 

 are measured. In the following analysis the origin from which 

 the angles <j> lf <f> 3 , <f> 5 , etc., are measured is determined by the selec- 

 tion of the initial ordinate (see Fig. 2) where y = y when x = o. 



It is convenient, when plotting, to measure time or angle from 

 the zero of the fundamental wave. We therefore rewrite (2) 

 by substituting x ^ for x\ thus, 



^3 A Y. V /_ & 



^sin^sin^ . 3 , v . , v s 



or, 



y = C l sin x + C 3 sin $(x -f 3 ) + C 6 sin 5(.r + a 5 ) (4) 



where 



<, 0, 



r 3 JL . '5 J x-.4.-^ / \ 



a = d> ot = rf> etc. (4a) 



"4 C > ' 



* (2b). In computing 0, note the signs of B and A; thus 



with + B and + ^4, we have <f> = + tan' 1 ; 



73 



with B and /4, we have <}>= + tan" 1 - 180 ; 



A 



Tt 



with B and +^4, we have 0= tan' 1 



A 



n 



with + B and /4, we have 0= tan' 1 -- 180 ; 



yi 



where -f tan" 1 _ is a positive angle (from o to +9) an( l tan " 1 ^ 

 negative angle (from o to 90). 



