Different expressions for group velocity

In summary, the conversation discusses various expressions for group velocity, including g = dw/dk and g = v+k dv/dk. The question is raised about the physical meaning of v in the latter equation, as it is not typically considered a characteristic velocity of a wave packet.
  • #1
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Homework Statement



Ok so we have different expressions for group velocity, g...

The most obvious being g = dw/dk

But then the text I am using says sinve w=vk we can write

g = v+k dv/dk

alongside other expressions, including k etc..

My question is - what is v in the equation above? I thought there were only two characteristic velocities of a wave packet which is the superposition of many different waves - the group and phase velocities.. What is the physical meaning of v here..

Thanks
 
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  • #2
Compare the equation [itex]\omega=vk[/itex] to the definition of phase velocity and you will see:wink:
 

1. What is group velocity and how is it different from phase velocity?

Group velocity is the velocity at which the envelope of a wave packet propagates through a medium. It is different from phase velocity, which is the velocity at which the phase of a wave travels. In other words, group velocity refers to the motion of the overall shape of a wave while phase velocity refers to the motion of individual wave crests or troughs.

2. What are some other terms used to describe group velocity?

Other terms used to describe group velocity include signal velocity, energy velocity, and velocity of propagation.

3. How is group velocity related to the dispersion of a wave?

The group velocity of a wave is inversely proportional to the rate of change of its frequency with respect to its wavenumber. In other words, the more a wave's frequency changes as its wavenumber increases, the lower its group velocity will be.

4. Can group velocity be greater than the speed of light?

Yes, in certain circumstances, it is possible for the group velocity of a wave to be greater than the speed of light. This is known as superluminal group velocity and is often observed in materials with negative refractive indices.

5. How does group velocity affect the behavior of a wave in a medium?

The group velocity of a wave can affect its dispersion, as well as the amount of energy and information that can be transmitted through a medium. It can also impact the behavior of waves in the presence of other waves, such as in wave interference phenomena.

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