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 wavelength (lambda) and time period as Equivalently, in terms of the wave's angular frequency , which specifies angular change per unit of time, and wavenumber (or angular wave number) , which represent the angular change per unit of space, To gain some basic intuition for this equation, we consider a propagating (cosine) wave . We want to see how fast a particular phase of the wave travels. For example, we can choose , the phase of the first crest. This implies , and so . Formally, we let the phase and see immediately that and . So, it immediately follows that As a result we observe a inverse relation between the angular frequency and wavevector. If the wave has higher frequency oscillations, the wave length must be shortened for the phase velocity to remain constant. 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. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin. Group velocity The group velocity of a collection of waves is defined as 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 with their respective angular frequencies and wavevectors. So, we have a product of two waves: an envelope wave formed by and a carrier wave formed by . We call the velocity of the envelope wave the group velocity. We see that the phase velocity of is In the continuous differential case, this becomes the definition of the group velocity. Refractive index In the context of electromagnetics and optics, the frequency is some function of the wave number,