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I have read the standard explanations on this, but I still have trouble with convincing myself that information is not propagating at faster than the speed c in media of refractive index less then unity (where c is the speed of light in the vacuum). I run into problems when I try to think about it mathematically. Here my query relates to phase velocity. I understand the argument that a given wave speed corresponds to monochromatic light and that this implies an infinitely long sinusoidal wave and thus no information is transmitted. But I have had a thought experiment that naively seems to conflict with that argument and I need help understanding where I am going wrong. Consider a wave pulse of the form:

$f(x,t)=\int_{-\infty}^{\infty}g\left (k \right )e^{i\left ( kx-\omega(k) t \right )}dk$

(equation 1)

And let us say at time t=0, we were to play God and somehow we created a light pulse exactly one full period of a sinusoidal wave:

$f(x,0)=1+cos(kx)$

between $-\frac{\pi}{k} \leq x \leq \frac{\pi}{k}$

and

$f(x,0)=0$

for $x <-\frac{\pi}{k}$ and $ \frac{\pi}{k} < x$

Mathematically we should be able to do this as we have:

$f(x,0)=\int_{-\infty}^{\infty}g\left (k \right )e^{i\left ( kx \right )}dk$

and we should be able to find $g(k)$ by taking the Fourier transform of $f(x,0)$

The point is that this is a localised pulse. And next we then let this pulse propagate in time t. According to equation 1 we effectively have the sum of infinitely wide pure monochromatic waves, where in each case , their speed of propagation is determined by:

$v_{p}=\omega(k)/k$

where k is the wave number and $\omega$ is the angular frequency which depends on k. And for at least some range of k we have $v_{p}\geq c$

My problem is this:

We can state that each wave is monochromatic and infinitely wide and thus carries no information. But we started with localised pulse, and this pulse will begin moving in the +x direction (albeit dispersed). Then at some point $ X > \frac{\pi}{k}$, and a certain time T we will find that

$f(X,T) \neq 0$

in a time $T < \frac{X-\frac{\pi}{k}}{c}$

So isn't information actually being propagated? And isn't some of it propagating at speeds > c ?

Where am I getting it wrong? Maybe the answer could be that it is physically nonsense to construct a wave in this way. Also I realize that in reality there are two field $B(x,t)$ and $E(x,t)$, and not a scalar function $f(x,t)$

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A localized pulse does not propagate at the phase velocity, $v_p=\omega/k$, but at the group velocity, $v_g=\frac{d\omega}{dk}$.

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  • $\begingroup$ Hi. Thanks for that. I was aware of the $v_{g}$, but I never thought of it for localised pulses. But it makes sense. Can you answer a related question for me? Is it the case still that actual information can only propagate at speed c through the medium? In the wave analysis I only see two wave speeds, $v_{p}$ and $v_{g}$ as you have just described. I cant see a wave speed = c anywhere. Is it the case that for refractive index < 1 materials that $v_{g} \leq c$ $\endgroup$ – Rory Cornish Jun 16 at 10:47
  • $\begingroup$ @RoryCornish, For example, in communications systems information is transmitted using pulses, and these propagate at the group velocity, $v_g \le c$. Also, in dispersive media, pulses or wave packets are distorted, and attenuated if the medium introduces loss. Details depend on the specific dispersion relation $\omega(k)$. $\endgroup$ – amateurAstro Jun 16 at 14:20
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It is only the velocity of light in space that cannot be surpassed. In a medium it can happen:

Cherenkov radiation (pronunciation: /tʃɛrɛnˈkɔv/) is an electromagnetic radiation emitted when a charged particle (such as an electron) passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. The characteristic blue glow of an underwater nuclear reactor is due to Cherenkov radiation.

In classical physics when the velocity of sound in a medium is exceeded by the velocity of a projectile one gets the sonic boom.

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  • $\begingroup$ Phase velocity of gamma waves in water or most glasses exceeds vacuum light speed. $\endgroup$ – Poutnik Jun 16 at 6:51

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