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Waveform and Spectrum

Waveform and Spectrum. A visual Fourier Analysis. String with fixed ends. …including 10 harmonics. …including 100 harmonics. Wave form. Sin(2  f t) + Sin(2  2f t) + Sin(2  3f t) +…. How about the amplitude?. A 1 Sin(2  f t) + A 2 Sin(2  2f t) + A 3 Sin(2  3f t) +….

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Waveform and Spectrum

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  1. Waveform and Spectrum A visual Fourier Analysis

  2. String with fixed ends

  3. …including 10 harmonics

  4. …including 100 harmonics

  5. Wave form Sin(2 f t) + Sin(2 2f t) + Sin(2 3f t) +… How about the amplitude? A1 Sin(2 f t) + A2Sin(2 2f t) +A3Sin(2 3f t) +… Does every harmonic contribute the same? How does the wave form change if we vary the Amplitude for each harmonic?

  6. From wave form to spectrum… A1 Sin(2 f t) + A2Sin(2 2f t) +A3Sin(2 3f t) +… Amplitude frequency f 2f 3f 4f 5f

  7. Amplitude Relative Amplitude Time frequency …back to wave form 50 harmonics 5 harmonics

  8. Influence of Phase (/2 for each) 3f, shifted by 2/3λ f 2f 2f, shifted by /4

  9. Influence of Phase (/2 for each) 10 harmonics 3 harmonics 50 harmonics

  10. Fourier Analysis • Joseph Fourier (1768-1830) Any periodic vibration can be build from a series of simple vibrations whose frequencies are harmonics of a fundamental frequency, by choosing the proper amplitude and phase.

  11. Applets for Fourier transformation • http://falstad.com/fourier/ • http://www.phy.ntnu.edu.tw/java/sound/sound.html • http://www.colorado.edu/physics/2000/applets/fourier.html

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