Thephase velocity of awave is the rate at which the wavepropagates in any medium. This is thevelocity at which the phase of any onefrequency component of the wave travels. For such a component, any given phase of the wave (for example, thecrest) will appear to travel at the phase velocity. The phase velocity is given in terms of thewavelengthλ (lambda) andtime periodT as

${\displaystyle v_{\mathrm{p}}=\frac{\lambda }{T}.}$

Equivalently, in terms of the wave'sangular frequencyω, which specifies angular change per unit of time, andwavenumber (or angular wave number)k, which represent the angular change per unit of space,

${\displaystyle v_{\mathrm{p}}=\frac{\omega }{k}.}$

To gain some basic intuition for this equation, we consider a propagating (cosine) waveA cos(kx −ωt). We want to see how fast a particular phase of the wave travels. For example, we can choosekx -ωt = 0, the phase of the first crest. This implieskx = ωt, and sov =x /t =ω /k.

Formally, we let the phaseφ =kx -ωt and see immediately that ω = -dφ / dt andk = dφ / dx. So, it immediately follows that

${\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }t}=-\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }t}\frac{\mathrm{\partial }x}{\mathrm{\partial }\varphi }=\frac{\omega }{k}.}$

As a result, we observe an inverse relation between the angular frequency andwavevector. If the wave has higher frequency oscillations, thewavelength must be shortened for the phase velocity to remain constant.^[2] Additionally, the phase velocity ofelectromagnetic radiation may – under certain circumstances (for exampleanomalous dispersion) – exceed thespeed of light in vacuum, but this does not indicate anysuperluminal information or energy transfer.^{[citation needed]} It was theoretically described by physicists such asArnold Sommerfeld andLéon Brillouin.

The previous definition of phase velocity has been demonstrated for an isolated wave. However, such a definition can be extended to a beat of waves, or to a signal composed of multiple waves. For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier. Thus the phase velocity of the carrier determines the phase velocity of the wave set.^[3]

Thegroup velocity of a collection of waves is defined as

${\displaystyle v_g=\frac{\mathrm{\partial }\omega }{\mathrm{\partial }k}.}$ 

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

 

So, we have a product of two waves: an envelope wave formed byf₁ and a carrier wave formed byf₂. We call the velocity of the envelope wave the group velocity. We see that thephase velocity off₁ is

${\displaystyle \frac{{\omega }_2-{\omega }_1}{k_2-k_1}.}$ 

In the continuous differential case, this becomes the definition of the group velocity.

In the context of electromagnetics and optics, the frequency is some functionω(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 lightc and the phase velocityv_p is known as therefractive index,n =c /v_p =ck /ω.

In this way, we can obtain another form for group velocity for electromagnetics. Writingn =n(ω), a quick way to derive this form is to observe

${\displaystyle k=\frac{1}{c}\omega n(\omega )⟹dk=\frac{1}{c}\left(n(\omega )+\omega \frac{\mathrm{\partial }}{\mathrm{\partial }\omega }n(\omega )\right)d\omega .}$ 

We can then rearrange the above to obtain

${\displaystyle v_g=\frac{\mathrm{\partial }w}{\mathrm{\partial }k}=\frac{c}{n+\omega \frac{\mathrm{\partial }n}{\mathrm{\partial }\omega }}.}$ 

From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency$\mathrm{\partial }n/\mathrm{\partial }\omega =0$ . When this occurs, the medium is called non-dispersive, as opposed todispersive, where various properties of the medium depend on the frequencyω. The relation${\displaystyle \omega (k)}$  is known as thedispersion relation of the medium.