COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

232 chapter 10y(t)10Y( )10–1t–10 20Figure 10.1 Left: A sawtooth function in time. Right: The Fourier spectrum of frequencies innatural units contained in this sawtooth function.10.2 Fourier Series (Math)Part of our interest in nonlinear oscillations arises from their lack of study intraditional physics courses even though linear oscillations are just the first approximationto a naturally oscillatory system. If the force on a particle is always towardits equilibrium position (a restoring force), then the resulting motion will be periodicbut not necessarily harmonic. A good example is the motion in a highly anharmonicwell p ≃ 10 that produces an x(t) looking like a series of pyramids; this motion isperiodic but not harmonic.In numerical analysis there really is no distinction between a Fourier integraland a Fourier series because the integral is always approximated as a finite series.We will illustrate both methods. In a sense, our approach is the inverse of thetraditional one in which the fundamental oscillation is determined analytically andthe higher-frequency overtones are determined by perturbation theory [L&L,M 76].We start with the full (numerical) periodic solution and then decompose it intowhat may be called harmonics. When we speak of fundamentals, overtones, andharmonics, we speak of solutions to the linear boundary-value problem, for example,of waves on a plucked violin string. In this latter case, and when given the correctconditions (enough musical skill), it is possible to excite individual harmonics orsums of them in the seriesy(t)=b 0 sin ω 0 t + b 1 sin 2ω 0 t + ··· . (10.4)Anharmonic oscillators vibrate at a single frequency (which may vary with amplitude)but not with a sinusoidal waveform. Expanding the oscillations in a Fourierseries does not imply that the individual harmonics can be excited (played).You may recall from classical mechanics that the general solution for a vibratingsystem can be expressed as the sum of the normal modes of that system. These−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 232