What is phase velocity and how does it relate to group velocity?

[mycotheology]

According to the wiki page, this diagram:

shows the phase velocity and group velocity moving in opposite directions. I think I understand what group velocity is, its the velocity of the overall wavepacket, in the diagram I can see a Gaussian wavepacket moving to the right so I'm guessing that's the group velocity. I can't see the phase velocity. I don't know what I'm looking for because I don't really know what phase means. I understand the concept of constructive and destructive interference and how 2 waves can be in or out of phase but I don't understand what phase means with respect to an individual wave. Does the word phase mean where the waveform is on the y axis, at a particular point on the x axis? Thats the idea I have of phase and with that in mind, I can't understand how phase can have a velocity.

 

[jtbell]

According to the wiki page, this diagram:

shows the phase velocity and group velocity moving in opposite directions. I think I understand what group velocity is, its the velocity of the overall wavepacket, in the diagram I can see a Gaussian wavepacket moving to the right so I'm guessing that's the group velocity.

Correct!

I can't see the phase velocity.

The phase velocity refers to the individual waves "inside" the wave packet, which are moving to the left in your example. Actually, what you're seeing is the average phase velocity of the individual waves that combine to produce the wave packet.

 

[Simon Bridge]

I think the way to see the phase motion is to pick an individual peak and keep your finger on it.

 

[mycotheology]

The phase velocity refers to the individual waves "inside" the wave packet, which are moving to the left in your example. Actually, what you're seeing is the average phase velocity of the individual waves that combine to produce the wave packet.

So if I did a Fourier transform on the wavepacket and broke it down into its constituent sine waves, they'd be moving to the left. Still can't see it though. Are you able to see the individual waves inside the packet in that diagram?

I think the way to see the phase motion is to pick an individual peak and keep your finger on it.

Whats confusing me is the way each individual ripple is growing and shrinking as it moves from left to right. In other words, each peak is moving up and down the y-axis as it traverses the x axis. I thought that phase depended on the position on the y axis. If I follow an individual peak, I do see it moving from right to left though, should I just forget about the y axis?

 

[arildno]

Group velocity is, essentially, the velocity by which ENERGY propagates. Energy is what effectively determines the actual AMPLITUDE of a given peak, whereas the peak itself moves with a different velocity. THAT velocity is what we call phase velocity.

 

[mycotheology]

Ignore the up and down motion. Just identify a peak - any peak - and consider its left-right velocity. That is the phase velocity.

Ah right, that clears it up, thanks a lot.

 

[Simon Bridge]

That's it :)
The phase is the angle that the phasor makes with the horizontal axis.
If the length of the phasor is constant, then that corresponds to the y-coordinate.
But if the length of the phasor is a function of time, then the same phase points may have different y-positions. All the peaks have the same phase - and they are easy to see - so...

http://en.wikipedia.org/wiki/Phasor (second diagram)