What's the Difference Between Phrase Velocity and Omega?

[asdf1]

phrase velocity and "omega"

what's the difference between phrase velocity and "omega"?

 

[jtbell]

Phrase velocity depends on how fast you can talk.

No wait, you're asking about wave equations like this one, right?

$$y = A \sin (kx - \omega t)$$

$\omega$ is the angular frequency of the wave, in radians per second. It's related to the usual frequency (cycles per second) by the number of radians per cycle: $\omega = 2 \pi f$. Both describe the rate at which any particular point on the wave oscillates up and down (or back and forth, or whatever).

Phase velocity (not "phrase velocity") is how fast a "crest" of the wave moves in the direction the the wave is traveling in. You can calculate it either as period/wavelength (think distance/time), or as $\omega / k$.

 

[asdf1]

i see... then what's the difference between phase velocity and group velocity?
they seem similar~

 

[jtbell]

Try this animation:

http://www.phys.virginia.edu/classes/109N/more_stuff/Applets/sines/GroupVelocity.html

The individual waves in the group move at the phase velocity. The shape of the group as a whole (the "envelope" of the individual waves) moves at the group velocity.

In the initial settings for this animation, the phase velocity is smaller than the group velocity, so the individual waves appear to be moving backwards with respect to the groups, although they're actually moving forwards in an absolute sense.

 

[asdf1]

thank you very much! :)

 

[Gokul43201]

The important difference is that while the phase velocity is given by $\omega/k$, the group velocity is $d \omega /dk$. Both are determined from the dispersion relation $\omega(k)$

To bring out the contrast, in certain nanosized magnetic materials, it is possible to have spin-wave modes (magnons), where the phase velocity is opposite in direction to the group velocity. The dispersion relation is a positive-valued curve with a negative slope.

See this animation (the last of the 6 files) :

http://www.csupomona.edu/~ajm/materials/animations/packets.html

 

[asdf1]

thanks! :)