Group Velocity: Estimating for Differing Wavenumbers

[hanson]

I am wondering if group velocity can defined only for waves with very close wavenumber?
I see a number of simulations which shows the superposition of two waves with slighlty different wavenumber and angular frequency, and a train of wave pulses is produced. The group velocity is then (w2-w1)/(k2-k1).

But what if it is two waves of very different wavenumber?

And also, for waves that is composed of a range of values of k, where shall I evluate the group velocity i.e. the derivative dw/dt at?

 

[jtbell]

I think the concept of "group velocity" has meaning only when \omega is a linear function of k, or nearly so. Then the derivative d\omega / dk is a constant, or nearly so, and it doesn't make any difference what value of k you evaluate the derivative at.

If d\omega / dk isn't constant, the packet changes shape as time passes, and eventually "falls apart".

 

[hanson]

I think the concept of "group velocity" has meaning only when \omega is a linear function of k, or nearly so. Then the derivative d\omega / dk is a constant, or nearly so, and it doesn't make any difference what value of k you evaluate the derivative at.

If d\omega / dk isn't constant, the packet changes shape as time passes, and eventually "falls apart".

If dw/dk is not constant, so what is the group velocity at every instant time? Though the packet changes shape as time passes, it shall still have a group velocity, isn't it? Just that the shape of the envelop changes but not the group velocity...?

And I actually see some books having Cg = 1-k^-2 etc...so...what does it mean?

 

[hanson]

I am thinking about the following:
That group velocity is for a group of waves that have a range of wavenumbers that is closed together (or dominated by such a group of wavenumbers) so that
k0 - delta < k <k0 +delta, where the dominating range of wavenumbers is around k0.

So, the group of waves can be viewed as a single harmonic wave of wavenumber k0 but with a varying envelope A(x,t). Just like the case of "beats".

So, the group velocity is dw/dk evaluated at k=k0.

So no matter the wave is dispersive or non-dispersive, the group velocity is still k=k0. But for dispersive waves, the envelop function will spread out or flatten as time passes.

Do you think I am thinking correctly? Is the concepts of group velocity has meaning only when the wavenumbers are dominated by a small group of wavenumbers / the wavenumbers are close enough together?

 

[Claude Bile]

We can still define a group velocity, one just has to be mindful that group velocity will vary with frequency in a medium, due to material dispersion. Dispersive behaviour is contained within higher order derivative terms, d^2w/dk^2 and so forth.

Claude.

 

[hanson]

We can still define a group velocity, one just has to be mindful that group velocity will vary with frequency in a medium, due to material dispersion. Dispersive behaviour is contained within higher order derivative terms, d^2w/dk^2 and so forth.

Claude.

Suppose the group velocity cg = 1-k^-2. And there is a group of waves with different values of k, so what value of k shall I use to find the group velocity?

 

[Meir Achuz]

Pick a reasonable central value. If the spread in k is so large that vg would be very different for different choices, then group velocity is not a useful concept.
The packet would spread so much that a packet velocity would be hard to specify.