Frequency dispersion in groups of
gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the
group velocity. In this deepwater case, the phase velocity is twice the group velocity. The red dot overtakes two green dots when moving from the left to the right of the figure.
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.
For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.
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

v_\mathrm{p} = \frac{\lambda}{T}.
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) ν_{p}:

v_\mathrm{p} = \frac{\omega}{k}.
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.
Relation to group velocity, refractive index and transmission speed
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,

\cos[(k\Delta k)x(\omega\Delta\omega)t]\; +\; \cos[(k+\Delta k)x(\omega+\Delta\omega)t] = 2\; \cos(\Delta kx\Delta\omega t)\; \cos(kx\omega t),
the amplitude becomes a sinusoid with phase speed 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.^{[2]}
In a given medium, the frequency is some function ω(k) of the wave number, so in general, the phase velocity v_{p} = ω/k and the group velocity v_{g} = dω/dk depend on the frequency and on the medium. The ratio between the phase speed v_{p} and the speed of light c is known as the refractive index, n = c/v_{p} = ck/ω. Taking the derivative of ω = ck/n with respect to k, we recover 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 the group speed is equal to the phase speed only when the refractive index is a constant: dn/dk = 0, and in this case the phase speed and group speed are independent of frequency: ω/k=dω/dk=c/n. ^{[2]} Otherwise, both the phase velocity and the group velocity vary with frequency, and the medium is called dispersive; the relation ω=ω(k) is known as the dispersion relation of the medium.
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.
See also
References

^ 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.

^ ^{}a ^{b} ^{c} "Phase, Group, and Signal Velocity". Mathpages.com. Retrieved 20110724.
Other


Main, Iain G. (1988), Vibrations and Waves in Physics (2nd ed.), New York: Cambridge University Press, pp. 214–216,

Tipler, Paul A.; Llewellyn, Ralph A. (2003), Modern Physics (4th ed.), New York: W. H. Freeman and Company, pp. 222–223,
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.
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