The phase velocity of a wave is the rate at which the phase of the wave propagates in space. 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 λ (lambda) and period T as
Or, equivalently, in terms of the wave's angular frequency ω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) νp:
To understand where this equation comes from, imagine a basic sine wave, A cos (kx−ωt). Given time t, the source produces ωt/2π = ft oscillations. At the same time, the initial wave front propagates away from the source through the space to the distance x to fit the same amount of oscillations, kx = ωt. So that the propagation velocity v is v = x/t = ω/k. The wave propagates faster when higher frequency oscillations are distributed less densely in space.2 Formally, Φ = kx−ωt is the phase. Since ω = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.
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 vg = Δω/Δk. 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.2 In reality, the vp = ω/k and vg = dω/dk ratios are determined by the media. The relation between phase speed, vp, and speed of light, c, is known as refractive index, n = c/vp = ck/ω. Taking the derivative of ω = ck/n, we get the group speed,
Noting that c/n = vp, this shows that group speed is equal to phase speed only when the refractive index is a constant: dn/dk = 0.2 Otherwise, when the phase velocity varies with frequency, velocities differ and the medium is called dispersive and the function, , from which the group velocity is derived is known as a dispersion relation. The group velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – 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.
- Cherenkov radiation
- Dispersion (optics)
- Group velocity
- Propagation delay
- Shear wave splitting
- Wave propagation
- Wave propagation speed
- Planck constant
- Matter wave#Phase velocity
- Nemirovsky, Jonathan; Rechtsman, Mikael C and Segev, Mordechai (9 April 2012). "Negative radiation pressure and negative effective refractive index via dielectric birefringence". Optics Express 20 (8): 8907–8914. doi:10.1364/OE.20.008907.
- "Phase, Group, and Signal Velocity". Mathpages.com. Retrieved 2011-07-24.
- Brillouin, Léon (1960), Wave Propagation And Group Velocity, New York and London: Academic Press Inc., ISBN 0-12-134968-3
- Main, Iain G. (1988), Vibrations and Waves in Physics (2nd ed.), New York: Cambridge University Press, pp. 214–216, ISBN 0-521-27846-5
- Tipler, Paul A.; Llewellyn, Ralph A. (2003), Modern Physics (4th ed.), New York: W. H. Freeman and Company, pp. 222–223, ISBN 0-7167-4345-0
- Subluminal – A Java applet
- Simulation – A Java applet by Paul Falstad
- Group and Phase Velocity – Java applet showing the difference between group and phase velocity.
|Velocities of waves|
|Phase velocity • Group velocity • Front velocity • Signal velocity|