Relation between Group velocity and Phase velocity

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Discussion Overview

The discussion revolves around the relationship between group velocity and phase velocity, exploring various methods to derive this relationship. Participants mention Fourier transforms and the conditions under which group velocity is defined, focusing on theoretical aspects and implications in wave mechanics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants inquire about unconventional methods to establish the relationship between group and phase velocities, mentioning Fourier transforms as a standard approach.
  • It is noted that phase and group velocities are meaningful primarily in the context of Fourier transforms, with specific references to the variables involved.
  • One participant emphasizes that for group velocity to be meaningful, the wave packet must consist of wavenumbers within a narrow range, highlighting a condition necessary for proper interpretation.
  • A later reply reiterates the importance of the narrow wavenumber range for meaningful group velocity, mentioning that the equation v_g=dk/dw arises from a Taylor expansion and critiques certain papers for overlooking this condition.

Areas of Agreement / Disagreement

Participants express agreement on the necessity of a narrow wavenumber range for meaningful group velocity, but there is no consensus on the methods to derive the relationship between group and phase velocities, indicating multiple competing views.

Contextual Notes

The discussion highlights the dependence on Fourier transforms and the assumptions regarding wave packet composition, which are not fully resolved.

neelakash
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Is there any unconventional method to find out the relation between group velocity and phase velocity? know there is a method employing Fourier tarnsforms and another easier method as well
 
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Phase and group velocities only have meaning using a Fourier transform.
w/k and dk/dw are in terms of Fourier variables.
 
one thing to notice is that, for a group velocity to be meaningful, the wave packet has to be merely composed of wavenumber in a narrow range.
 
Fizik said:
one thing to notice is that, for a group velocity to be meaningful, the wave packet has to be merely composed of wavenumber in a narrow range.
That is a very good point. The equation v_g=dk/dw comes from keeping only the first term of a Taylor expansion. Many of the papers claiming funny group velocities for light miss the point you mention.
 

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