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? 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) +…

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

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

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

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

Fourier Analysis  Joseph Fourier ( ) 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.

Applets for Fourier transformation   tml tml  ourier.html ourier.html