User:Stigmatella aurantiaca/sandbox

The phase velocity of a wave is the rate at which any one frequency component of the wave travels. On the other hand, the group velocity of a wave is the rate with which modulations of the wave's amplitudes travel through space.

In the above figure, the red dot moves with the phase velocity, following the crest of one particular frequency component of the wave. The green dots move at the rate of modulations in the wave. In this figure, note that the phase velocity is double the group velocity. Source: https://en.wikipedia.org/wiki/File:Wave_group.gif

v_{\mathrm {p} }={\frac {\lambda }{T}}={\frac {\omega }{k}}

The phase velocity equals the wavelength $\lambda$ divided by the period $T;$ alternatively, phase velocity equals the angular frequency $\omega$ divided by the angular wave number $k.$

Consider a solution to the wave equation for a pure sinusoidal traveling wave:

y(x,t)=\cos {(kx-\omega t)}

A pure sine wave does not transmit any information. In order to transmit information, some form of modulation must be added to the carrier tone, some change in amplitude or frequency. Consider the sum of two waves of slightly differing frequency $\Delta \omega$ and wave number $\Delta k$ :

y(x,t)=\cos[(k-\Delta k)x-(\omega -\Delta \omega )t]\;+

\cos[(k+\Delta k)x-(\omega +\Delta \omega )t]=

2\cos(\Delta kx-\Delta \omega t)\;\cos(kx-\omega t)