This movie is an animation of a wave packet that is a sine wave modulated by a Gaussian envelope. Because the wave is not spread uniformly throughout space, it consists of a spread of wavelengths. In such a situation, there are two significant velocities:
- The phase velocity: vp = ω/k is the speed of a point of constant phase (e.g., one of the wave crests).
- The group velocity: vg = dω/dk is the speed of the envelope. I won't derive it here.
See this page.
This prescription only makes sense if the spread of wavelengths is small enough that we don't need to worry about where (at what ω) to evaluate the velocities. In other words, we can ignore the second derivative.
Consider an electromagnetic wave. In vacuum, ω = ck, so both the phase and group velocities = c. However, in a material, the index of refraction, n, is usually a function, n(ω), of frequency, so ω = [c/n(ω)]k is no longer a simple relationship. We now have:
- vp = ω/k = c/n
- vg = dω/dk = vp/[1+(ω/n)(dn/dω)]
Depending on whether n is increasing or decreasing with ω, vp can be larger or smaller than vg. In the movie, vp is larger. Crests appear at the trailing (left) edge of the packet, catch up, then disappear out the front (right) edge.
In the video, the packet is not spreading out even though vg ≠ vp. Spreading requires a non-zero second derivative. In that case, vg depends on ω. and the high frequency part of the envelope has a different speed than the low frequency part.
Jon Thaler
Dec. 2013