Derivation of group velocity.

In summary, group velocity is a concept in wave mechanics that refers to the speed at which the overall shape or envelope of a wave packet propagates through a medium. It is different from phase velocity, which only takes into account the speed of individual wave crests or troughs. The mathematical derivation of group velocity involves using the wave equation and taking the derivative with respect to time. Group velocity is significant in understanding wave behavior in different mediums and has practical applications in optics, telecommunications, and quantum mechanics.
  • #1
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I was reading the derivation on Wikipedia:

http://en.wikipedia.org/wiki/Group_velocity#Derivation

Why is the first part before the integral sign ignored when calculating the velocity? Surely it would also cause a phase shift in some time interval and make the waves move forward (or backward)?
 
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  • #2
Group velocity cares about the speed of the amplitude, phases are not relevant for the group velocity.
 
  • #3
Ok, but when calculating the velocity by dividing the terms in front of t by those in front of k, why can some of the terms in front of t be ignored?
 

What is group velocity?

Group velocity is a concept in wave mechanics that refers to the speed at which the overall shape or envelope of a wave packet (a group of waves) propagates through a medium.

How is group velocity different from phase velocity?

Group velocity is different from phase velocity in that it takes into account the speed at which the overall shape or envelope of a wave packet travels, while phase velocity refers to the speed at which an individual wave crest or trough travels through a medium.

What is the mathematical derivation of group velocity?

The mathematical derivation of group velocity involves using the wave equation, which describes the relationship between the frequency, wavelength, and speed of a wave, and taking the derivative with respect to time. This results in an equation for group velocity that takes into account the dispersion of the wave packet.

What is the significance of group velocity in wave mechanics?

Group velocity is significant because it helps us understand how wave packets behave in different mediums. It can also help us predict the behavior of waves in complex systems, such as in optics or quantum mechanics.

How is group velocity used in practical applications?

Group velocity is used in a variety of practical applications, including in optics for designing lenses and other optical devices, in telecommunications for signal transmission, and in quantum mechanics for understanding the behavior of particles at the quantum level.

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