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? Does every harmonic contribute the same? How does the wave form change if we vary the Amplitude for each harmonic? A 1 Sin(2 f t) + A 2 Sin(2 2f t) +A 3 Sin(2 3f t) +…

6. From wave form to spectrum… A 1 Sin(2 f t) + A 2 Sin(2 2f t) +A 3 Sin(2 3f t) +… Amplitude frequency f 2f3f4f5f

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

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

9. Influence of Phase (  /2 for each) 3 harmonics 10 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://falstad.com/fourier/  http://www.phy.ntnu.edu.tw/java/sound/sound.h tml http://www.phy.ntnu.edu.tw/java/sound/sound.h tml  http://www.colorado.edu/physics/2000/applets/f ourier.html http://www.colorado.edu/physics/2000/applets/f ourier.html