Question

Why is wave packet adequate for the transmission of signals?

Answer 1

A Pure travelling wave is given by the equation,


 
By modulation or wave mixing techniques we can superimpose several travelling waves of different parameters- frequencies, amplitudes and phases .Some sources of signals in nature itself is in superimposed form. For example –we know light is a wave and White light is mixture of lights of different frequencies in the electromagnetic spectrum. Signals we obtain during earth quake (seismic waves ) or ECG machines ,etc –all have superimposed  waves. This superimposed  wave package is called as ‘ Wave Train’ or ‘Wave envelope’ or   ‘Wave pulse’ . Now the velocity with which the entire ‘wave envelope ‘  moves is the ‘group velocity’.

Whereas each component wave of this ‘wave envelope ‘ travels at its own speed known as the phase velocity (v= wavelength/time period).

In principle any wave envelope  can be represented by a periodic function. We use Fourier analysis methods to mathematically decompose a wave function into its harmonics–sine and cosine components .

Now by the term ‘Information’ we refer to the entire shape of the ‘wave envelope’ –that is frequencies, amplitudes and phases of all the component waves that superimpose to give this wave envelope. Consider a source launching a wave pulse that passes through a non-dispersive or normally dispersive medium and reaching a detector. Now If the detector tries to reconstruct the wave envelope by setting sampling rate based on phase velocity, then it will end up recording only few of the component waves of the wave envelope. Because of this it cannot reconstruct the wave envelope exactly. (in signal processing -this is called as aliasing )
Thus, it can be seen ‘information’ in this context can only be transmitted at group velocity.
 
Localized wave packet are also wave envelopes that are usually considered to be short bursts ( like photons) –(i.e) interfere constructively only  in small limited region, and cancel out everywhere else. Producing a localized wave packet requires continuous distribution of frequencies (i.e) frequency spectrum must be continuous. The wave envelope should have countably infinite component waves. But anyway since it is also a wave envelope and  any dispersive medium has negligible overall effect because of so many  component waves, ‘Information’ travels at group velocity of Localized wave packet.


 



graphic above shows building of wave envelope adding one wave component at a time and below is shown the transmission of wave envelope -one can see how its shape changes within a cycle itself.Thus  only if we move a window over it at the 'group velocity'-we will be able to capture all the 'Information' about it shape (amplitude variation within a cycle,phase changes in a cycle ,etc). Whereas if we move a window at a phase velocity ,we can see only the parameters of few of its component waves.

( taken &  modified from  Uncertainty principle - infact there is also a way to explain 'Uncertainity ' based on fourier transform and the concepts we see here )


 
 
In case of propagation of light within a  normal dispersive medium both amplitude and frequency of individual component waves changes as refractive index  is dependent on frequency .Thus  group velocity is always slower than the phase velocity in a normal dispersive medium. In an anomalous dispersive medium, the group velocity is not only faster than the phase velocity it could also exceed the speed of light in vacuum. Nevertheless component waves (i.e) phase velocities never exceed velocity of light respecting special theory of relativity. In case of anomalous dispersive medium, one cannot claim Information is transmitted at group velocity as ‘Rephasing’ happens during transmission and therefore it is not causal. This ‘Rephasing’ of the wave components causes  Wave envelope to travel at superluminal speed ( greater than speed of light )
 
Does a group velocity larger than c violate relativity?

(I think you got  this question by reading David J Griffiths ?   ...also note wavelet is a seperately different concept )


thanks