The phase velocity can be faster than light. Some argue that the phase velocity doesn't convey information, but this doesn't convince me.
1, 1 Hz means 0. We emit single-frequency wave for a second. Then we don't emit anything for a second. And so on.The transitions between two frequencies will not move with the phase velocity! Nor will the beginning of transmission!
Only in the steady plane wave parts in the middle, the wave crests will move faster than light. And that's just "appearance" in the sense that bewteen any two times, you can find a maximum of the electric field and pretend the wave moved there. But actually, that's just our interpretation (finding a pattern in the "image"). It's like watching a movie on your screen - you interpret the change of a certain pattern in the image as motion, but actually the pixels don't move at all - they just change intensity so you perceive it as motion. That's all phase velocity tells you: you put in a single sine wave in and observe how fast the pattern goes. But actual propagation of momentum and energy (which is percieved most easily as the speed of the beginning of the beam) is slower than speed of light in the vacuum. What you'll see when you turn it on, is wave crests moving faster than the front, and "disappearing" in the front of the beam.
Point a laser to the sky, track where it lands very far away from the Earth, and then move your arm. The point you are tracking in space is moving faster than the speed of light, even though light itself isn't.
This isn't the same as group velocity, but it is very similar. There is no problem in having arbitrary points you imagine going faster than light, and relativity doesn't disallow it, as long as energy doesn't travel faster than it.
You have to be careful to interpret your statements 1 and 2 correctly.
In statement 1 you are talking about an infinitely long wave train which is made up of only one single frequency.
As soon as you have a wave train of finite length then the wave train is made up of the sum many frequencies.
So in statement 2 what you think of as a pulse of a wave with a frequency 2 Hz is actually a pulse of the superposition of very, very many frequencies which all add up to look like a pulse of a wave with frequency 2 Hz.
This subtle difference does not matter as long as the medium is no dispersive, ie the speed of the wave does not depend on its frequency. If the medium is dispersive then the different frequency components that make up your 2 Hz pulse travel at different speeds and the shape of you pulse changes. As the shape of you pulse changes which part of it do you use to measure its speed?
I think that conceptually this is very difficult.
There are many animations on the Internet which try and show visually what happens.
Here are a couple:
Link 1
Link 2 which gives you more control.
There are many others and I would be glad to hear of any ones which are better.
See Greg Egan's applet. This will clear up all confusion, and ought to be viewed by any reporter who covers a "scientists break light speed" report.
Subluminal shows how a wave composed of a multitude of frequencies moving at different velocities — all less than or equal to c, the speed of light in a vacuum — can appear to have features moving faster than c.
The grid that crosses the screen is moving with a velocity of c, and no individual frequency outpaces it. However, the total wave (the bottom trace, in white) has its strongest peaks where all the individual frequencies are in phase, and the places where that happens shift with time, at a “speed” that is greater than c. Nothing is actually travelling with these peaks, though; they’re just an artifact of the way the different frequencies are slipping in and out of phase.
This illusion of superluminal motion can only occur when the refractive index of the medium falls as the frequency of the light increases, a situation known as anomalous dispersion. If it falls rapidly enough, the group velocity — the speed at which the overall envelope of the wave seems to move — can even become negative.
If we take your case of amplitude modulating two frequencies with pulses which are a second wide - then these are not pure frequencies any more - they will have a bandwidth of approx 2Hz (depends on pulse shape) [because there will be two sidebands at +/- 1Hz from carrier]. These pulse therefore have a group velocity.
Only if you emit a wave at a single frequency for all time would it have no group velocity. There would be no information content in such a signal of course.
