CHAPTER XL 

 WAVE ANALYSIS. 



EXPERIMENT n-A. Analysis of a Complex Wave by the 

 Method of 18 Ordinates. 



i. Introductory. An alternating current or electromotive 

 force is rarely an exact sine wave ; in addition to the fundamental 

 wave or first harmonic it usually comprises odd harmonics of 

 3> 5> 7> e *c., times the fundamental frequency. (Even harmonics* 

 are never present when the negative half-wave is a repetition of 

 the positive half-wave.) If we are given the ordinates or 

 certain ordinates of a complex wave, we can " analyze " it into 

 its components, that is, we can find the fundamental and the 

 harmonics of higher frequency of which it is composed, each 

 component wave being defined by its amplitude and its phase 

 position with respect to the fundamental. 



2. Any complex wave in which there are no even harmonics 

 (the negative half-wave being a repetition of the positive) can 

 be represented by a Fourier's series consisting of the following 

 sine and cosine terms : 



y = AI sin x + A 3 sin 3* + A 5 sin 5* 



+ B! COSJT + B 3 cos 3* + B 5 cos 5* . (i) 



a* is an angle varying with time; thus x = u>t, $x = $vt, etc., where 

 w is 2ir times the fundamental frequency. 



By combiningf the sine and cosine terms, (i) may be written 



* For the analysis of waves with even harmonics, see Appendix II. 

 t(2a). To prove equation (2), expand as follows: 



V^ + 5 Z sin (* + 0) = VA 2 + B 2 (cos sin *-f- sin cos #) 



= A sin x -f- B cos x, 

 where 



A = VA 2 + B 2 cos ; B = VA 2 + B 2 sin ; B-r-A = tan 0. 

 This will be seen more clearly by constructing a right triangle with C as 

 the hypothenuse and A and B as the two sides. 



