Insensitivity of the human ear to phase.

The human ear is insensitive to phase in the Fourier expansion of a periodic waveform: given a periodic signal [math]A_0/2+\sum\limits_{n=0}^{+\infty}A_n\cos(nt+\phi_n)[/math], we can replace it with [math]A_0/2+\sum\limits_{n=0}^{+\infty}A_n\cos(nt+\tilde\phi_n)[/math], where we the phase angles [math]\phi_n[/math] are replaced with other randomly chosen numbers [math]\tilde\phi_n[/math]. The two waveforms are possibly very different, but sound exactly the same!

Sound only with the java version of the applet: in case the java plugin doesn't work in the browser, please open the ggb file in your local copy of GeoGebra.