Is Group Velocity Merely a Mathematical Concept Compared to Phase Velocity?

[neelakash]

Homework Statement

I do not know if it is a homework problem or not.The moderators will determine that.However,I read in acoustics books that it is a standard practice to use superposition of harmonic functions to denote a finite wave train, which cannot be given by a sine/cosine functions(because they are infinitely extended in time and space).This wave tran is known as wave group.
The impression I got is that a wave group is a mathematical concept and have no physical reality.Yet,its velocity, i.e. group velocity is different fro the phase velocity(w/k) of the wave.
So,where we are?It seems that phase velocity (w/k) is the physical reality but the group velocity is a mathematical concept only.However,phase velocity may be >c,whereas group velocity always<c.Please make the misconception clear.

Homework Equations

The Attempt at a Solution

 

[mjsd]

group velocity is the speed of actual information propagation and that's why it must be bounded by c.
if you have just one wave, you can't transmit information, but if you have say two waves then, you can transmit information by adjusting the phase between the two waves accordingly. In that case, the phase difference "encodes" your information, and it travels at the group velocity, not the phase velocity (which is the velocity of the individual wave...determined by the frequency and wave length). same principle works when you have many waves put together. mathematically, group velocity is given by dw/dk (the derivative clear indicating the "difference" between two things)

 

[m2003]

Phase velocity and group velocity are both important concepts in wave propagation, particularly in acoustics and other areas of physics. While they are both related to the speed at which a wave travels, they represent different aspects of the wave.

Phase velocity is the speed at which the phase of a wave travels, meaning the location of the peaks and troughs of the wave. It is given by the formula v = ω/k, where ω is the angular frequency and k is the wavenumber. This velocity can exceed the speed of light in certain cases, but this does not violate the laws of physics as it is not the speed at which energy or information is transmitted.

Group velocity, on the other hand, is the speed at which the envelope of a wave group travels. A wave group is a localized disturbance that is made up of many different frequencies, which can be represented by the superposition of harmonic functions. The group velocity is given by the formula v_g = dω/dk, where ω is the angular frequency and k is the wavenumber. This velocity represents the speed at which energy or information is transmitted through the wave.

While phase velocity and group velocity may have different values, they both have physical significance. Phase velocity tells us about the speed of the individual components of a wave, while group velocity tells us about the overall propagation of the wave. It is possible for the phase velocity to be greater than the speed of light in certain situations, but this does not violate the laws of physics as it is not the speed at which energy or information is transmitted.

In summary, phase velocity and group velocity are both important concepts in wave propagation and should not be seen as competing or contradictory. They represent different aspects of the same wave and both have physical significance in understanding the behavior of waves. It is important to understand the distinction between the two and to use them appropriately in different contexts.