What is "the frequency" of a wavepacket?

Question

I'm making a numerical simulation of a transmission line, based on the Telegrapher's equations. And in particular of how pulses move through a cable. Currently I'm trying to understand how dispersion works but I ran into a problem. In any textbook on the topic of waves you will find the following formula for the group velocity: $$v_g(\omega) = \frac{\partial \omega}{\partial \beta}$$ Where beta is the complex part of the propagation constant gamma $$\gamma (\omega) = \alpha (\omega) + j\beta (\omega)$$ At the same time it is explained that the group velocity is 'the velocity of the envelop around the wavepacket'. What I don't understand is the following: What does omega mean exactly in this context of wavepackets? A wavepacket doesn't have a frequency (it's a collection of frequencies), but it does have a group velocity. But how can I calculate it's group velocity if I only have a formula which requires a frequency?

I tried to find the answer online but I could't figure it out. One term that pops up is the 'carrier frequency', but I'm not sure if that's the frequency I'm looking for. On another place I read it's the 'average frequency' but I'm not really sure what that means exactly and if it's true. If someone knows the answer to my problem it would make me very happy!

Answer 1

In your equation, ${v_g(\omega)}$ can be interpreted as "the speed at which the envelope of a wavepacket which is well localized in frequency space and has central frequency $\omega$ propagates".

To clarify what this means:

As you noted, a wavepacket doesn't contain one frequency. It will contain a range of frequencies typically centered around some central frequency $\omega_0$. If our signal is roughly Guassian, then the frequencies contained within the packet will also be approximately Gaussian and hence centred relatively closely around the central frequency. We can then approximate the wavepacket as having the single central frequency of the Guassian. This is actually a decent approximation in a broad range of settings.

The group velocity of this central frequency, $v_g(\omega_0)$, is then a good approximation of the speed at which the envelope of this signal propagates.

This is the context in which the group velocity is most meaningful - when there arent "too many frequencies" contained within the wave packet.

If there are a broad range of frequencies in the wavepacket (say between $\omega_+$ & $\omega_-$), there will likely be a substantial difference between the group velocity of the leading and trailing frequencies within the wavepacket. This means $v_g(\omega_+)$ and $v_g(\omega_-)$ will be quite different and the leading and trailing edge of the envelop will travel at different speeds. The envelope "disperses" and the group velocity is a less useful concept since we can no longer approximate it as being localized in frequency space.

Hope this helps.