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Group velocity as a concept in Classical Waves confuse me. It's very easy to point out visually, like in this really helpful graphic here. Okay, it's the speed of the moving bulge, which, notably moves opposite to the phase velocity.

I see what it looks like quite clearly, but there are key things about it I still don't understand.

  1. What are situations where the graphic shown can describe a physical thing? What kind of waves have this property and why is it useful?

  2. Mathematically, group velocity is described as $$v_{g} = \frac{d \omega}{dk}$$

Or perhaps more loosely, the rate of change of angular frequency as a function of wavenumber. However, there is no correlation in my mind between the graphic and this equation. How can I relate my intuition to the mathematics?

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With a continuous wave you cannot transmit a signal. For a signal to be transmitted, you need a modulation of the wave, e.g. amplitude modulation. For example, to transmit acoustic frequencies (speech), you modulate the high frequency electromagnetic carrier wave (on the order of MHz for medium wave transmitters) with the acoustic frequencies(up to 20kHz). This modulation produces small variations called side-bands (plus and minus 20kHz) in the transmitted waves. The group velocity of a wave describes the velocity with which such modulation of the carrier amplitude, which transmits the signal, propagates. In free space, the group velocity of an EM wave is identical to the phase velocity $c$ because the dispersion is linear $\omega=c k$. Thus also a pulse shaped modulation propagates with unchanged form. On transmission lines, there can be significant nonlinear dispersion, i.e. the phase velocity $v_{ph}= \frac {\omega}{k}$ for different frequencies is not constant and, in general, different from the group velocity $v_{gr}=\frac {\partial \omega}{\partial k}$. This leads to a loss of shape of a pulse-like modulation of the carrier wave. However, the propagation speed of such a pulse modulation can still be obtained from the group velocity.

That the group velocity is opposite to the phase velocity happens only in systems with special nonlinear dispersion relations.

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  • $\begingroup$ I wonder if a tangible analogy is that you can't send smoke signals, with a only constant stream of smoke - you have to create an alternation between clear air and smoke, in order to signal (even if the lighting of the fire in the first place, is the only alternation that occurs - that is enough to send a simple signal to anyone who can see that the state of the system has changed from clear air to smoked air). $\endgroup$ – Steve Feb 4 '18 at 18:51
  • $\begingroup$ I'm afraid I'm having trouble understanding this. First, I'll try and wrap my head around the first sentence. Why is a continuous wave unable to serve as a signal? With @Steve 's analogy, are you saying that a constant stream of smoke cannot serve as a signal because it has no variation of form? You need to periodically turn off and on the signal to produce a message? Kind of like morse code? $\endgroup$ – sangstar Feb 4 '18 at 18:56
  • $\begingroup$ @sangstar, yes. You can't send morse code, if all you hear is either constant silence or constant tone - it's the alternation between them that sends a signal. A fire alarm would send no signal, if it sounded constantly at all times (or did not sound at all under any circumstances). $\endgroup$ – Steve Feb 4 '18 at 18:59
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    $\begingroup$ It was also the fate that befell the boy who cried wolf - his cries ceased to signal that a wolf was present, because he cried whether or not the wolf was present. $\endgroup$ – Steve Feb 4 '18 at 19:01
  • $\begingroup$ @Steve with you there then. Now, with sound then, for instance, it's clear that a continuous wave could be heard but wouldn't be able to send any signal for the same logic as we've just agreed upon. How then, would this modulation then, solve things? I'm afraid the answer hasn't helped me immediately grasp what this does to the continuous wave to send a signal. $\endgroup$ – sangstar Feb 4 '18 at 19:03
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When we describe phonons we using the dispersion relation (angular velocity vs wavevector). And the slope of this graph can tell you how fast this phonons move (group velocity).

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