HABMONIC ANALYSIS. 



77 



is that shown in the harmonic plot in figure 74. A 'straight line or even 

 a dot may be represented by the results of a harmonic analysis, but an 

 infinite number of terms is required. 



The harmonic analysis is a purely mathematical operation. Applied 

 to a vibration it shows— provided it is sufficiently extended— how that 

 vibration might have been produced, but not how it actually was produced. 

 When a curve has been produced by adding a number of simple sinusoids 

 with harmonic periods, the analysis will give as results the true ampU- 



tudes of the components actually used and for 



the amphtudes of all other members. What will 



be the results of analysis when simple sinusoids 



have been used for the composition 



whose periods do not belong in the 



harmonic series? 



Curves whose periods do not 

 coincide with any members of the 

 harmonic series are said to be 

 "inharmonic." It is important to 

 inquire how an inharmonic sinu- 

 soid will show itself in the results 

 of a harmonic analysis. The result 

 is utterly different from that for a 

 harmonic. For example, the sinu- 

 soid with the period ^T, appears 

 in the result only as a sinusoid 

 with the period ^T; it does not 

 appear at all in the other members 

 of the series. Its harmonic plot 

 shows a single ordinate. The sinu- 

 soid, however, with the period '^ T, having no place of its own in the 

 results, appears in every member, more strongly in members whose periods 

 are near its own, less strongly in members with periods that differ more. 

 Such a simple sinusoid (figure 75) would appear in a harmonic analysis 

 into 36 members as shown in figure 76. 



It is important to remember that an inharmonic in the midst of a 

 number of harmonics as components of a curve appears in the results 

 of a harmonic analysis only as an intensification of the harmonics. A 

 curve composed of a series of sinusoids with the periods T, ^T, ^T, {T, 

 \T, and the amphtudes 10, ,5, 5, 5, 5 (actual plot in figure 77), gives, 

 when analyzed, a harmonic plot which is identical with the actual plot. 

 The harmonic analysis of a curve composed of simple sinusoids with the 



Fig. 71. — Curve with its sinu- 

 soid components. 



12 3 4 

 Fig. 72.— Har- 

 monic plot to 

 figure 71. 



