Waves/Fourier Transforms

Fourier Transform So far, you've learned how to superimpose a finite number of sinusoidal waves. However, a wave in general can't be expressed as the sum of a finite number of sines and cosines. Fortunately, we have a theorem called Fourier's theorem which basically states that under certain technical assumptions, any function, f(x) is equal to an integral over sines and cosines. In other words, . Now, if we're given the wave function when t=0, φ(x,0) and the velocity of each sine wave as a function of its wave number, v(k), then we can compute φ(x,t) for any t by taking the inverse Fourier transform of φ(x,0) conducting a phase shift, and then taking the Fourier transform. Fortunately, the inverse Fourier transform is very similar to the Fourier transform itself. This tells us that, since waves which are very spread out, like the sine wave, have a narrow range of wave numbers, wave functions whose wave numbers are very spread out will only be significant at a narrow range of positions. Fourier Transform Properties Fourier Transform Pairs Further reading Engineering Analysis/Multi-Dimensional Fourier Series The Feyman lectures Chapter 50 Harmonics.