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Chapter 6: Q11CQ (page 223)

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.

Short Answer

Expert verified

The values of phase and group velocities depend on the central value of k under consideration.

Step by step solution

01

Definition of group and phase velocities

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.

02

Explanation and conclusion

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.

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