Expressions for the group velocity

In summary, the group velocity in physics is the velocity at which the overall shape or envelope of a wave packet propagates. It can be calculated by taking the derivative of the dispersion relation and is often equal to the phase velocity, except in cases of dispersion. Some real-world applications of group velocity include fiber optics and radar/sonar systems. According to the theory of relativity, the group velocity cannot exceed the speed of light, but in certain materials with negative refractive index, it can appear to be greater. The group velocity can affect the behavior of waves in various ways, such as causing dispersion in media and playing a role in quantum mechanics.
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Homework Statement


Okay so I've been deriving various expressions for the group velocity starting with

g=dw/dk

then g=v+kdv/dk

also g=v-lambda dv/dlambda (*)

then finally g=c/u(1+lambda/u (du/dlambda))

Ok so finally I am asked to use equn (*) to derive the following expression

g=v[1-1/(1+(v/lambda')(d lambda'/dv))

where lambda' is the wavlength in vacuum..


not sure how to do it..any help would be great thanks

Homework Equations





The Attempt at a Solution

 
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FAQ: Expressions for the group velocity

What is the group velocity in physics?

The group velocity in physics is the velocity at which the overall shape or envelope of a wave packet propagates. This is in contrast to the phase velocity, which is the velocity at which the individual wave crests or troughs propagate.

How is the group velocity calculated?

The group velocity can be calculated by taking the derivative of the dispersion relation, which relates the frequency and wavevector of a wave. In most cases, the group velocity will be equal to the phase velocity, but in cases of dispersion, they may differ.

What are some real-world applications of group velocity?

One example of a real-world application of group velocity is in fiber optics, where the group velocity is used to determine the speed at which information can be transmitted through a fiber optic cable. Group velocity is also used in radar and sonar systems to determine the speed of moving objects.

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

No, according to the theory of relativity, the group velocity, like the phase velocity, cannot exceed the speed of light. However, in certain materials with negative refractive index, the group velocity can appear to be greater than the speed of light, but this is a result of the way the wave is propagating through the material and does not violate the laws of relativity.

How does group velocity affect the behavior of waves?

The group velocity can affect the behavior of waves in a number of ways. For example, in dispersive media, the group velocity can cause different frequencies of a wave to travel at different speeds, resulting in the separation of colors in a prism. In quantum mechanics, the group velocity plays a role in the behavior of particles, such as in the phenomenon of wave-particle duality.

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