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Visual Fourier Analysis: From Waveform to Spectrum in String with Fixed Ends

Explore the visual Fourier analysis of a string with fixed ends, including 10 and 100 harmonics. Understand harmonic contributions to the waveform and how changing amplitudes affect it. Delve into amplitude, frequency, time, and influence of phase on the waveform. Learn from Joseph Fourier's theory and find Fourier analysis applets.

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Visual Fourier Analysis: From Waveform to Spectrum in String with Fixed Ends

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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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