Editing Phase velocity (section)

== Group velocity ==

[[File:Wavepacket1.gif|thumb|right|A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line).]]

The [[group velocity]] of a collection of waves is defined as

:<math> v_g = \frac{\partial \omega} {\partial k}.</math>

When multiple sinusoidal waves are propagating together, the resultant superposition of the waves can result in an "envelope" wave as well as a "carrier" wave that lies inside the envelope. This commonly appears in wireless communication when [[modulation]] (a change in amplitude and/or phase) is employed to send data. To gain some intuition for this definition, we consider a superposition of (cosine) waves {{mvar|f(x, t)}} with their respective angular frequencies and wavevectors.

:<math>\begin{align}
f(x, t) &= \cos(k_1 x - \omega_1 t) + \cos(k_2 x - \omega_2 t)\\
&= 2\cos\left(\frac{(k_2-k_1)x-(\omega_2-\omega_1)t}{2}\right)\cos\left(\frac{(k_2+k_1)x-(\omega_2+\omega_1)t}{2}\right)\\
&= 2f_1(x,t)f_2(x,t).
\end{align}</math>

So, we have a product of two waves: an envelope wave formed by {{math| ''f''<sub>1</sub> }} and a carrier wave formed by {{math| ''f''<sub>2</sub> }}. We call the velocity of the envelope wave the group velocity. We see that the '''phase velocity''' of {{math| ''f''<sub>1</sub> }} is  
:<math> \frac{\omega_2 - \omega_1}{k_2-k_1}.</math>
In the continuous differential case, this becomes the definition of the group velocity.