GREPhysics.NET: GR9277 #79

GR9277 #79

Problem

GREPhysics.NET Official Solution

\prob{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

They are in opposite directions.
They are in the same direction and the phase velocity is larger
They are in the same direction and the group velocity is larger
The phase velocity is infinite and the group velocity is finite.
They are the same in direction and magnitude

Wave Phenomena $\Rightarrow$ }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).

Alternate Solutions

calcuttj
2014-09-05 06:30:22

You can instantly eliminate D.

If you don't remember specifically which is $\partial\omega$ / $\partial$ k 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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calcuttj
2014-09-05 06:30:22

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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?

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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.

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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?

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