Verification of of Phase vs. Group Velocity

In summary, to verify the statement that the phase velocity and group velocity are equal when there is no dispersion, you can use the formula Phase Velocity = w/k and Group Velocity = dw/dk. If w/k is constant, the dispersion relation will be a straight line with a gradient of w/k, making the group velocity equal to w/k for all w. This can be seen by comparing the local slope of the curve to the slope of the line connecting any two points on it. Graphing w vs k can help visualize this concept.
  • #1
CollectiveRocker
137
0
How do i go about verifying this statement: If the phase velocity is the same for all wavelenghts of a certain wave phenomenon (no dispersion), the group and phase velocities are equal. Do i just put the two formulas equal to each other and solve for something or what?
 
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  • #2
any ideas on how to even start?
 
  • #3
Phase Velocity = w/k

Group Velocity = dw/dk

If w/k is constant, it really is not that hard to show that the phase and group velocities are equal.

Claude.
 
  • #4
Explain please.
 
  • #5
Look at the dispersion relation (w vs k). If w/k is constant, the dispersion relation is just a straight line with gradient w/k.

Since the group velocity is the gradient, dw/dk, if the dispersion relation is a straight line, then the group velocity will be w/k for all w.

Claude.
 
  • #6
I think he be suggesting that the local slope of a curve is be equal to the slope of the line joining any two points on it, if the curve is a straight line.
 
  • #7
Spot on. Graphing w vs k may make it less confusing.

Claude.
 

1. What is the difference between phase velocity and group velocity?

Phase velocity refers to the speed at which a specific phase of a wave travels, while group velocity refers to the speed at which the overall energy of a wave travels.

2. How do you calculate phase velocity and group velocity?

Phase velocity can be calculated by dividing the wavelength by the period of the wave, while group velocity can be calculated using the partial derivative of the wave's dispersion relation.

3. Why is verifying phase velocity and group velocity important in scientific research?

Verifying phase and group velocity allows scientists to accurately understand the properties of different waves and their behavior, which is crucial in various fields such as optics, acoustics, and quantum mechanics.

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

No, according to the theory of relativity, the speed of light is the maximum speed at which any form of energy can travel. Therefore, phase velocity cannot exceed the speed of light.

5. How does the medium through which a wave travels affect its phase and group velocity?

The phase and group velocity of a wave are influenced by the properties of the medium through which it travels, such as density, elasticity, and temperature. Different mediums can cause changes in the speed and direction of the wave, leading to variations in phase and group velocity.

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