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,


 * $$f(x)=\int_{-\infty}^{\infty}(c_1(k)\cos(kx)+c_2(k)\sin(kx)) dk$$.

Now, if we're given the wave function when t=0, &phi;(x,0) and the velocity of each sine wave as a function of its wave number, v(k), then we can compute &phi;(x,t) for any t by taking the inverse Fourier transform of &phi;(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.


 * $$c_1(k)=\frac{1}{2\pi}\int_{-\infty}^{\infty} f(x)\cos (kx)\,dx \quad

c_2(k)=\frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) \sin (kx)\,dx $$

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.