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\prob{79}
9277_79

The dispersion curve shown above relates the angular frequency $\omega$ to the wave number k. For waves with wave numbers lying in the range $k_1


  1. They are in opposite directions.
  2. They are in the same direction and the phase velocity is larger
  3. They are in the same direction and the group velocity is larger
  4. The phase velocity is infinite and the group velocity is finite.
  5. They are the same in direction and magnitude

Wave Phenomena}Group Velocity

Recall that the group velocity is given by v_g = \frac{d\omega}{dk} and the phase velocity is given by v_p = \omega/k.

In the region between k_1 and k_2, the derivative is a constant negative quantity (approximately just the derivative of a line with negative slope). However, \omega/k is positive in this region. Thus, the phase and group velocity are traveling in opposite directions. Thus, choose choice (A).

See below for user comments and alternate solutions! See below for user comments and alternate solutions!
Alternate Solutions
calcuttj
2014-09-05 06:30:22
You can instantly eliminate D.

If you don't remember specifically which is \partial\omega/\partialk and which is \omega/k, just remember that one depends on the derivative which is negative and the other is w/k which is positive and you can come to A.
Alternate Solution - Unverified
Comments
calcuttj
2014-09-05 06:30:22
You can instantly eliminate D.

If you don't remember specifically which is \partial\omega/\partialk and which is \omega/k, just remember that one depends on the derivative which is negative and the other is w/k which is positive and you can come to A.
Alternate Solution - Unverified
hoyas08
2007-12-30 12:08:56
The second sentence of the question doesn't display right for me. It should read:

For waves with wave numbers lying in the range k₁ < k < k₂ , which of the following is true of the phase velocity and the group velocity?
NEC
jcain6
2005-11-23 06:46:20
Choice (c) states that they are in the same direction and the group velocity is larger. This isn't true and just looks to be a typo on your part. Your description is consistent with choice (A) which is the correct answer.

yosun
2005-11-23 14:59:43
jcain6: thanks for the typo-alert; it has been corrected.
Fixed Typos!

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You can instantly eliminate D. If you don't remember specifically which is \partial\omega/\partialk and which is \omega/k, just remember that one depends on the derivative which is negative and the other is w/k which is positive and you can come to A.

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