1. Fourier analysis Periodic function: Any (“reasonable”) periodic function, can be written as a series of sines and cosines “vibrations”, whose frequencies are harmonics of a fundamental frequency, by choosing the proper amplitudes and phases of these harmonics. Compare this statement with the fact that practically any music can be played on piano. 1

2. Another definition of a 0 2

3. Some properties to review Kronecker delta: 3

4. Spectrum - specification of the strengths of the various harmonics Examples: A square wave has a spectrum with a fundamental followed by odd harmonics with the ratio of the amplitudes being 1/n A triangle wave has a spectrum with a fundamental followed by odd harmonics, but the ratio of the amplitudes is 1/n 2 A saw tooth wave has a spectrum with both odd and even harmonics and amplitude ratio of 1/n 2 f AnAn 1 3 5 7 Square wave 4

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7. 7 The RMS and Parseval’s theorem For definition on slide 1: For definition on slide 2:

8. 8 Fourier series solution for damped driven oscillator (Linear superposition) Resonance can be observed at different harmonics!