Some doubts on signal velocity

[issacnewton]

Hi

There are several posts here on group velocity, phase velocity and the confusion caused
by it. I also read that article posted on mathpages.com (who is the author ?). I have some
questions after reading these things. Like that article says, there three kinds of 'speeds'
concerning wave motion. Phase velocity , group velocity , signal velocity. Its only the
signal velocity which really matters because information travels at that speed. I have few
questions here. Why don't we see this distinction between several kinds of 'speeds' being mentioned in books like Young,Freedman or Serway, Jewett when they introduce
waves and their velocities. All the confusion about these matters will not be there if
introductory books talked with clarity.

Another thing the article says is about the signal velocity. We can derive the relationship between the phase and group velocities of a signal. Is there any way we can derive the
relationship between the signal velocity and other two velocities ?

thanks

 

[sophiecentaur]

For the simplest example of a continuous wave, there is no information on the wave so the only relevant velocity is the phase velocity - there is no signal and the group velocity would be the same as the phase velocity. This may be why the two textbookd don't discuss it.
Things only get more complicated when the wave is modulated in some way, carrying information. aamof, when the bandwidth of the information is a tiny fraction of the carrier frequency, there is no distinction either.

 

[issacnewton]

So is the treatment of group , phase and signal velocity more rigorous at advanced level books ? May be J.D.Jackson ?

I have seen on this forum, while replying to the questions on such matters, people say that relativity is not violated since the information doesn't travel at the speed greater than light. Einstein's postulates of STR don't talk in terms of "information" . So is there
any additional equivalent statement of the postulate or is there any mathematical theorem which follows from the postulates, which talks in terms of "information" .
I am sure there is mathematically rigorous way of talking about signal velocity and its
connection to "information" content

 

[sophiecentaur]

The group velocity is given by δω/δk and this is the speed at which information is carried - it is the speed of the envelope (in forms of modulation where there actually is an envelope).

 

[issacnewton]

its not true for anomalous dispersion

http://en.wikipedia.org/wiki/Anomalous_dispersion#Group_and_phase_velocity

 

[sophiecentaur]

Agreed. But with dispersion how does one define the time of arrival of the information? It becomes a more pragmatic business that would need to include SNR as well, I think.