Phase velocity

So, it immediately follows that As a result, we observe an inverse relation between the angular frequency and wavevector.

If the wave has higher frequency oscillations, the wavelength must be shortened for the phase velocity to remain constant.

[2] Additionally, the phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in vacuum, but this does not indicate any superluminal information or energy transfer.

[citation needed] It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin.

For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier.

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 f(x, t) with their respective angular frequencies and wavevectors.

Frequency dispersion in groups of gravity waves on the surface of deep water. The ■ red square moves with the phase velocity, and the ● green circles propagate with the group velocity . In this deep-water case, the phase velocity is twice the group velocity . The red square overtakes two green circles when moving from the left to the right of the figure.
New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.
For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.
Propagation of a wave packet demonstrating a phase velocity greater than the group velocity.
This shows a wave with the group velocity and phase velocity going in different directions. The group velocity is positive, while the phase velocity is negative. [ 1 ]
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).