Advanced Calculus fi..

Advanced Calculus, Fifth EditionFigure 7.1 Function with period 2n.If f (x) has period p, then the substitution:r*. Yurtx=p-~ T T3-e-P&/- . (7.4)converts f (x) into a function of t having period 2n; for when t increases by 2n, xincreases by p.A function f (x) having period 2n is illustrated in Fig. 7.1. Such periodic functionsappear in a great variety of physical problems: the vibrations of a spring; themotion of the planets about the sun; the rotation of the earth about its axis; themotion of a pendulum; the tides and wave motion in general; vibrations of a violinstring, of an air column (for example, in a flute); and musical sounds in general.The modern theory of light is based on "wave mechanics," with periodic vibrationsa characteristic feature; the spectrum of a molecule is simply a picture of the differentvibrations taking place simultaneously within it. Electric circuits involve manyperiodically varying variables, for example, the alternating current. The fact that ajourney around the globe involves a total change in longitude of 360" is an expressionof the fact that the rectangular coordinates of position on the globe are periodicfunctions of longitude, with period 360"; many other examples of such periodicfunctions of angular coordinates can be given.Now it can be shown that every periodic function of x satisfying certain verygeneral conditions can be represented in the form (7.1), that is, as a trigonometricseries (see Section 7.3). This mathematical theorem is a reflection of a physicalexperience most vividly illustrated in the case of sound, for example, that of a violinstring. The term ;ao represents the neutral position, the terms a, cos x + b1 sinx thefundamental tone, the terms a2 cos 2x + b2 sin 2x the first overtone (octave); the otherterms represent higher overtones. The variable x must here be thought of as time andthe function f (x) as the displacement of an instrument, such as a phonograph needle,which is recording the sound, or of a point on the string. Thus the musical tone heardis a combination of simple harmonic vibrations-the terms (a, cos nx + b, sin nx).Each such pair can be written in the formwhereA,=,/=, an=Ansincr, bn=Ancoso.The "amplitude" A,+% is a measure of the importance of the nth overtone in thewhole sound. The differences in the tones of different musical instruments can beascribed mainly to the differences in the weights A, of the overtones.