The diagram below plots ω(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.
The values of phase and group velocities depend on the central value of k under consideration.
Group velocity is defined as the velocity of the whole envelope of a wave in space.
Phase velocity is defined as the velocity of a phase or part of a wave in space.
For small (central) values of k, the slope of the curve, or tangent line, dω/dk is tany, and therefore the slope of a line from the origin, ω/k is relatively large. That is, the group velocity is smaller than the phase velocity. When the central value of k is large, these two quantities are equal, and so are the phase and group velocities.
Hence, the values of phase and group velocities depend on the central value of k under consideration.
Example 6.3 gives the refractive index for high-frequency electromagnetic radiation passing through Earth’s ionosphere. The constant , related to the so-called plasma frequency, varies with atmospheric conditions, but a typical value is . Given a GPS pulse of frequency traveling through of ionosphere, by how much, in meters, would the wave group and a particular wave crest be ahead of or behind (as the case may be) a pulse of light passing through the same distance of vacuum?
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