The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelengthλ (lambda) and time periodT as
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 without dispersion.
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.
Equivalently, in terms of the wave's angular frequencyω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represent the angular change per unit of space,
To gain some basic intuition for this equation, we consider a propagating (cosine) wave A cos(kx − ωt). We want to see how fast a particular phase of the wave travels. For example, we can choose kx - ωt = 0, the phase of the first crest. This implies kx = ωt, and so v = x / t = ω / k.
Formally, we let the phase φ = kx - ωt and see immediately that ω = -dφ / dt and k = dφ / dx. So, it immediately follows that
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).
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 communications, 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.
So, we have a product of two waves: an envelope wave formed by f1 and a carrier wave formed by f2. We call the velocity of the envelope wave the group velocity. We see that the phase velocity of f1 is
In the continuous differential case, this becomes the definition of the group velocity.
In the context of electromagnetics and optics, the frequency is some function ω(k) of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light c and the phase velocity vp is known as the refractive index, n = c / vp = ck / ω.
In this way, we can obtain another form for group velocity for electromagnetics. Writing n = n(ω), a quick way to derive this form is to observe
We can then rearrange the above to obtain
From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is a constant dn / dk = 0. When this occurs, the medium is called non-dispersive, as opposed to dispersive, where various properties of the medium depend on the frequency ω. The relation ω = ω(k) is known as the dispersion relation of the medium.