The phase velocity of a wave is the rate at which the wavepropagates in some 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 will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength and time period as Equivalently, in terms of the wave's angular frequency , which specifies angular change per unit of time, and wavenumber , which represents the proportionality between the angular frequency and the linear speed ν, To understand where this equation comes from, consider a basic cosine wave, . After time , the source has produced oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance to fit the same number of oscillations,. Thus the propagation velocity v is. The wave would have to propagate faster when higher frequency oscillations are distributed less densely in space unless the wave length is compensatorily shortened. Formally, is the phase, where Since and, the wave velocity is.
Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known as modulation, is required. By combining two sines with slightly different frequencies and wavelengths, the amplitude becomes a sinusoid with phase speed. It is this modulation that represents the signal content. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity, vg. In a given medium, the frequency is some function of the wave number, so in general, the phase velocity and the group velocity depend on the frequency and on the medium. The ratio between the speed of light c and the phase velocity vp is known as the refractive index,. Taking the derivative of with respect to, would yield the group velocity, except one cannot create a group with only a finite number of wave frequencies/wave vectors. Noting that, indicates that the group speed is equal to the phase speed only when the refractive index is a constant, and in this case the phase speed and group speed are independent of frequency,. Otherwise, both the phase velocity and the group velocity vary with frequency, and the medium is called dispersive; the relation is known as the dispersion relation of the medium. The group velocity of electromagnetic radiation may – under certain circumstances – exceed the speed of light in a 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. See dispersion for a full discussion of wave velocities; however, the article fails to address the failure by such researchers to realize that in such cases the so-called index of refraction disobeys fundamental assumptions about such an object so the "velocity" is not a group velocity at all.
