Rate at which the phase of the wave propagates in space
Frequency dispersion in groups ofgravity waves on the surface of deep water. The■ red square moves with the phase velocity, and the● green circles propagate with thegroup 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. 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.Propagation of awave packet demonstrating a phase velocity greater than the group velocity.This shows a wave with the group velocity and phase velocity going in different directions.[1] The group velocity is positive (i.e., theenvelope of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward).
Thephase velocity of awave is the speed of anywavefront, a surface of constantphase. This is thevelocity at which the phase of any constant-frequency component of the wave travels. For such aspectral component, any given phase of the wave (for example, thecrest) 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.[2]: 18
For a simple sinusoidal wave the phase velocity is given in terms of thewavelengthλ (lambda) andtime periodT as
Equivalently, in terms of the wave'sangular frequencyω, which specifies angular change per unit of time, andwavenumber (or angular wave number)k, which represent the angular change per unit of space,[2]
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.[3]
In the context of electromagnetics and optics, the frequency is some functionω(k) 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 lightc and the phase velocityvp is known as therefractive index,n =c /vp =ck /ω.
In this way, we can obtain another form for group velocity for electromagnetics. Writingn =n(ω), a quick way to derive this form is to observe
We can then rearrange the above to obtain
From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency. When this occurs, the medium is called non-dispersive, as opposed todispersive, where various properties of the medium depend on the frequencyω. The relation is known as thedispersion relation of the medium.
^abBorn, Max; Wolf, Emil (1993).Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (6 ed.). Oxford: Pergamon Press.ISBN978-0-08-026481-3.
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