Phase velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed 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

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

Or, equivalently, in terms of the wave's angular frequency ω, which is the number of oscillations per unit of time, and wavenumber k, which is the number of oscillations per unit of space, by

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

To understand where it comes from, imagine a basic sine wave, A cos (kx−ωt). Given time t, the source produces ωt 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 speed v is v = x/t = ω/k. The wave propagates faster when higher frequency oscillations are distributed less densely in space.^[1] Formally, Φ = kx−ωt is the phase. Since ω = −dΦ/dt and k = +dΦ/dx, the wave speed is v = dx/dt = ω/k.

1 Relation to group velocity, refractive index and transmission speed
2 Matter wave phase
3 Slow phase velocity and superluminality
4 See also
5 References
6 External links

Relation to group velocity, refractive index and transmission speed

Since 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,

$\cos[(k-\Delta k)x-(\omega-\Delta\omega)t]\; %2B\; \cos[(k%2B\Delta k)x-(\omega%2B\Delta\omega)t] = 2\; \cos(\Delta kx-\Delta\omega t)\; \cos(kx-\omega t),$

the amplitude becomes a sinusoid with phase speed of v_g = 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.^[1] In reality, the v_p = ω/k and v_g = dω/dk ratios are determined by the media. The relation between phase speed, v_p, and speed of light, c, is known as refractive index, n = c/v_p = ck/ω. Taking the derivative of ω = ck/n, we get the group speed,

$\frac{\text{d}\omega}{\text{d}k} = \frac{c}{n} - \frac{ck}{n^2}\cdot\frac{\text{d}n}{\text{d}k}.$

Noting that c/n = v_p, this shows that group speed is equal to phase speed only when the refractive index is a constant: dn/dk = 0.^[1] Otherwise, when the phase velocity varies with frequency, velocities differ and the medium is called dispersive. The phase 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.

Matter wave phase

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that

$v_\mathrm{p} = \frac{\omega}{k} = \frac{E/\hbar}{p/\hbar} = \frac{E}{p}.$

Using relativistic relations for energy and momentum, we have

$v_\mathrm{p} = \frac{E}{p} = \frac{\gamma m c^2}{\gamma m v} = \frac{c^2}{v} = \frac{c}{\beta}$

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, $\gamma$ the Lorentz factor, c the speed of light, and β the speed as a fraction of c. The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed $v < c$ for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.

$v_\mathrm{p} > c, \,$

and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, as it carries no information. See the article on signal velocity for details.

Slow phase velocity and superluminality

Contrarily, a subluminal phase velocity may be corresponding to a velocity faster than light, i.e. $v>c,$ because the product is constant $c^2$ .^[2]^[3] For example, the phase velocity of commercial electric power (frequency 60Hz) in copper is 3 m/s according to the original definition $v_\mathrm{p} = \frac{\omega}{k}$ . Thus, the velocity $v$ should be 10⁸ c on account of $v_\mathrm{p} = \frac{c^2}{v}.$ The value equals energy flow velocity $\mathbf{S}\over w \$ of the field given by classical electrodynamics ( $S$ is Poynting vector and $w$ is energy density).

References

^ ^a ^b ^c "Phase, Group, and Signal Velocity". Mathpages.com. http://www.mathpages.com/home/kmath210/kmath210.htm. Retrieved 2011-07-24.
^ Wang, Zhong-Yue (2011). "Superluminal energy transmission in the Goos-Hanchen shift of total reflection". Optics Communications 284 (7): 1747–1751. Bibcode 2011OptCo.284.1747W. doi:10.1016/j.optcom.2010.12.027.
^ Zhong-Yue Wang. Energy transport faster than light in good conductors. arXiv:/1101.1840.

Brillouin, Léon "Wave Propagation And Group Velocity" Academic Press Inc., New York and London (1960) ISBN 0-12-134968-3.
Main, Iain G. (1988).Vibrations and Waves in Physics. 2nd ed. New York; Cambridge University Press. ISBN 0-521-27846-5. 214-6 pp.
Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 222-3 pp.

External links

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