CHAPTER V. 

 HARMONIC ANALYSIS. 



When a well-made tuning-fork is set in vibration and a fine point at 

 the end of one of its prongs is drawn over a piece of glass coated with smoke, 

 it inscribes a curve of the form shown in figure 67. A curve of this charac- 

 ter is known as a " simple sinusoid." The extent of the elongation to each 

 side of the position of rest is called the "ampUtude" a; the time for one 

 complete vibration is the "period" T. The number of vibrations per 

 second is termed the "frequency"; the frequency is the reciprocal of the 

 period, or 71=1/7" (for example, if a fork has a period of r= 0.02s., its 

 frequency will be n= 1/0.02 = 50 per second). 



Some elementary facts concerning simple sinusoids must be borne 

 in mind; the tuning-fork vibration may be used to illustrate them. Since 

 the fork may vibrate more or less strongly, its amplitude may vary (figure 

 68). The record of the fork may begin at some point of the vibration not on 

 the axis. Figure 69 shows a record beginning at the extreme of positive 

 elongation. The condition of the vibration at any moment is known as 

 its " phase." The phase of the vibration in figure 69 is a quarter of a period 

 more advanced than that in figure 67. 



A series of vibrations whose periods have the relations I, h, h> • • • 

 (that is, with frequencies in the relations 1, 2, 3, . . . ) is termed a " har- 

 monic series." The amplitudes and phases in the series may have any 

 values. A harmonic series of simple sinusoids with the same amplitude 

 and phase at the start is shown in figure 70. 



According to Fourier's theorem* a curve of any form — not only a 

 complicated curve but even a straight line or a dot — may be expressed 

 as the sum of a harmonic series of simple sinusoids, provided the series is 

 sufficiently extended and the amplitudes and phases are properly adjusted. 

 A practical method of deducing the necessary sinusoids from the original 

 curve has been developed; the method is known as " simple harmonic analy- 

 sis." Briefly stated, a simple harmonic analysis is made in the following 

 way. A number — 12, 24, 36, 72 — of equidistant ordinates are measured 

 from the axis of the curve or from a line parallel to it. These ordinates 



♦Fourier, Th^orie Analytique de la Chaleur, Ch. Ill, Paris, 1822. For complete accountof work on 

 this and similar problems see Burkhardt, Entwickelungen nach oscillirenden Funktionen, Jahresbericht 

 d. deutschen Mathematiker-Vereinigung, 1901-02-03, x. 



76 



