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Visual Fourier Analysis of Waveform and Spectrum: Understanding Harmonics and Amplitude Relationships

Explore the influence of amplitude, frequency, and phase on waveforms and spectra using Fourier analysis. Study the impact of adjusting amplitudes and phases on harmonic components. Learn how to transform waveforms into spectra and vice versa. Discover the principles of Fourier analysis explained through interactive applets.

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Visual Fourier Analysis of Waveform and Spectrum: Understanding Harmonics and Amplitude Relationships

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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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