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Phase velocity

 
Sci-Tech Dictionary: phase velocity
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(′fāz və′läs·əd·ē)

(physics) The velocity of a point that moves with a wave at constant phase. Also known as celerity; phase speed; wave celerity; wave speed; wave velocity.

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Sci-Tech Encyclopedia: Phase velocity
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The velocity of propagation of a pure sine wave of infinite extent. In one dimension, for example, the form of the disturbance for such a wave is given by y(x, t) = A sin [2π(xλ − t/T)]. Here x is the position at which the disturbance y(x, t) exists at time t, λ is the wavelength, T is the period which is related to the wave frequency by T = 1/f, and A is the disturbance amplitude. The argument of the sine function is called the phase. The phase velocity is the speed with which a point of constant phase can be said to move. Thus x/λ − ft = constant, so the phase velocity vp is given by dx/dt = vp = λf. This is the basic relationship connecting phase velocity, wavelength, and frequency. See also Phase (periodic phenomena); Sine wave; Wave motion.

The phase velocity for waves in a medium is determined in part by intrinsic properties of the medium. For all mechanical waves in elastic media, the square of the phase velocity is proportional to the ratio of the appropriate elastic property of the medium to the appropriate inertia property. The phase velocity of electromagnetic waves depends upon the medium as well. In vacuum, the phase velocity c is given by c2 = l/∊0μ0 ≈ 9 × 1016 m2/s2, where ∊0 and μ0 are respectively the permittivity and permeability of the vacuum. Phase velocity may also depend upon the mode of wave propagation—in general, upon the frequency of the wave. Waves of different frequencies will travel at different speeds, resulting in a phenomenon called dispersion. See also Electromagnetic radiation; Light; Wave equation; Young's modulus.

Wikipedia: Phase velocity
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Phase velocity in periodic gravity waves on the surface of deep water. The red dot moves with the phase velocity, and is located at a fixed wave phase: the crest for the case shown.

The phase velocity (or phase speed) 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 speed 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 and wavenumber k by

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

In a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate.

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.

Contents

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, γ the Lorentz factor, c the speed of light, and β the velocity as a fraction of c. The variable v can either be taken to be the velocity of the particle or the group velocity of the corresponding matter wave. See the article on group velocity for more detail. Since the particle velocity v < c for any particle that has mass (according to special relativity) 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 velocity is in the relativistic range. The superluminal phase velocity does not violate special relativity, for it doesn't carry any information. See the article on signal velocity for detail.

See also

References

  • 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.
  • Léon Brillouin "Wave Propagation And Group Velocity" Academic Press Inc., New York and London (1960) ISBN 0-1213-4968-3.
  • Main, Iain G. (1988).Vibrations and Waves in Physics. 2nd ed. New York; Cambridge University Press. ISBN 0-5212-7846-5. 214-6 pp.

External links


Velocities of Waves 2006-01-14 Surface waves.jpg
Phase velocity | Group velocity | Front velocity | Signal velocity

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