Phase velocity

File:Wave group.gif

Frequency dispersion in groups of gravity waves on the surface of deep water. The Template:Colorbull red square moves with the phase velocity, and the Template:Colorbull green circles propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square overtakes two green circles when moving from the left to the right of the figure.Template:Paragraph New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.Template:Paragraph For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.

File:Wave packet propagation (phase faster than group, nondispersive).gif

Propagation of a wave packet demonstrating a phase velocity greater than the group velocity.

File:Wave opposite-group-phase-velocity.gif

This shows a wave with the group velocity and phase velocity going in different directions.<ref name=nemirovsky2012negative>Template:Cite journal</ref> The group velocity is positive (i.e., the envelope of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward).

The phase velocity of a wave is the speed of any wavefront, a surface of constant phase. This is the velocity at which the phase of any constant-frequency component of the wave travels. For such a spectral component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity of light waves is not a physically meaningful quantity and is not related to information transfer.<ref name=BornWolf-1993>Template:Cite book</ref>Template:Rp

Sinusoidal or plane waves

For a simple sinusoidal wave the phase velocity is given in terms of the wavelength Template:Mvar (lambda) and time period Template:Mvar as

<math>v_\mathrm{p} = \frac{\lambda}{T}.</math>

Equivalently, in terms of the wave's angular frequency Template:Mvar, which specifies angular change per unit of time, and wavenumber (or angular wave number) Template:Mvar, which represent the angular change per unit of space,<ref name=BornWolf-1993/>

<math>v_\mathrm{p} = \frac{\omega}{k}.</math>

Beats

The previous definition of phase velocity has been demonstrated for an isolated wave. However, such a definition can be extended to a beat of waves, or to a signal composed of multiple waves. For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier. Thus the phase velocity of the carrier determines the phase velocity of the wave set.<ref name="electroagenda">Template:Cite web</ref>

Dispersion

In the context of electromagnetics and optics, the frequency is some function Template:Math of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light c and the phase velocity v_p is known as the refractive index, Template:Math.

In this way, we can obtain another form for group velocity for electromagnetics. Writing Template:Math, a quick way to derive this form is to observe

<math> k = \frac{1}{c}\omega n(\omega) \implies dk = \frac{1}{c}\left(n(\omega) + \omega \frac{\partial}{\partial \omega}n(\omega)\right)d\omega.</math>

We can then rearrange the above to obtain

<math> v_g = \frac{\partial w}{\partial k} = \frac{c}{n+\omega\frac{\partial n}{\partial \omega}}.</math>

From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency <math display=inline>\partial n / \partial\omega = 0</math>. When this occurs, the medium is called non-dispersive, as opposed to dispersive, where various properties of the medium depend on the frequency Template:Mvar. The relation <math>\omega(k)</math> is known as the dispersion relation of the medium.

See also

• Cherenkov radiation
• Dispersion (optics)
• Group velocity
• Propagation delay
• Shear wave splitting
• Wave
• Velocity factor
• Planck constant
• Speed of light
• Matter wave#Phase velocity

References

Bibliography

• Crawford jr., Frank S. (1968). Waves (Berkeley Physics Course, Vol. 3), McGraw-Hill, Template:ISBN Free online version
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