Phase velocity and group velocity

In summary, the group velocity and phase velocity are two different concepts in physics. The group velocity, which is the same as the velocity of the particle, cannot exceed the speed of light according to the equations of special relativity. On the other hand, the phase velocity, which is the speed of the peaks of a wave, can theoretically travel faster than light. However, this does not violate the laws of physics as it is a mathematical concept and does not involve any actual physical objects traveling faster than light. This is further explained in the examples of a laser beam and a shadow projected onto a distant object. The concept of group velocity does not require quantum mechanics, but the concept of phase velocity does.
  • #1
BacalhauGT
21
1
I Still don't understand why the group velocity has to be less than c but phase velocity not.

Can you explain me this?

Thank you :cry:
 
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  • #2
The group velocity is the same as the velocity of the particle and can't be higher than the speed of light according to the equations of special relativity.

The phase velocity is the speed of the peaks of the wave and can travel faster than light. The speed of the peaks is more of a mathematical concept; no physical thing is traveling that fast. Since no energy or mass or information is going faster than light, the equations of special relativity are not violated.

It's related to these examples (copied from Wikipedia):
"If a laser is swept across a distant object, the spot of laser light can easily be made to move across the object at a speed greater than c. Similarly, a shadow projected onto a distant object can be made to move across the object faster than c. In neither case does the light travel from the source to the object faster than c, nor does any information travel faster than light."

I recently had a question related to this. The thread is here:
https://www.physicsforums.com/showthread.php?t=693357
Posts #8 and #9 have derivations of phase and group velocities that you might find helpful.
 
  • #3
waterfire said:
The group velocity is the same as the velocity of the particle and can't be higher than the speed of light according to the equations of special relativity.

The phase velocity is the speed of the peaks of the wave and can travel faster than light. The speed of the peaks is more of a mathematical concept; no physical thing is traveling that fast. Since no energy or mass or information is going faster than light, the equations of special relativity are not violated.

It's related to these examples (copied from Wikipedia):
"If a laser is swept across a distant object, the spot of laser light can easily be made to move across the object at a speed greater than c. Similarly, a shadow projected onto a distant object can be made to move across the object faster than c. In neither case does the light travel from the source to the object faster than c, nor does any information travel faster than light."

I recently had a question related to this. The thread is here:
https://www.physicsforums.com/showthread.php?t=693357
Posts #8 and #9 have derivations of phase and group velocities that you might find helpful.


When you say "The group velocity is the same as the velocity of the particle and can't be higher than the speed of light according to the equations of special relativity." you need quantum mechanics, right?
 
  • #4
BacalhauGT said:
When you say "The group velocity is the same as the velocity of the particle and can't be higher than the speed of light according to the equations of special relativity." you need quantum mechanics, right?

Group velocity is a property of waves. Without quantum mechanics a particle is not considered a wave so there is no group velocity or phase velocity, only the velocity of the particle.

You can say that the velocity of the particle can't be higher than the speed of light without quantum mechanics, but then there is no group velocity.
 

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

Phase velocity refers to the speed at which a single wave crest travels through a medium, while group velocity refers to the speed at which a group of waves, or a wave packet, travels through a medium. In other words, phase velocity describes the propagation of an individual wave, while group velocity describes the overall movement of a wave packet.

2. How are phase velocity and group velocity related?

Phase velocity and group velocity are related through the dispersion relation, which describes the relationship between a wave's frequency and its wavelength. The group velocity can be calculated by taking the derivative of the dispersion relation with respect to wavelength, while the phase velocity can be calculated by dividing the wave's frequency by its wavelength.

3. Can the phase velocity and group velocity be different?

Yes, the phase velocity and group velocity can be different in certain situations. This is known as "anomalous dispersion" and can occur when the medium has a non-linear response to the wave, or when the wave encounters a sharp boundary.

4. How do phase velocity and group velocity affect the behavior of waves?

The phase velocity and group velocity can affect the behavior of waves in different ways. For example, if the group velocity is faster than the phase velocity, the wave packet will spread out over time. Conversely, if the phase velocity is faster than the group velocity, the wave packet will become more compact.

5. Can phase velocity and group velocity be measured experimentally?

Yes, phase velocity and group velocity can be measured experimentally using various techniques such as interferometry or time-of-flight measurements. These measurements are important in understanding the properties and behavior of waves in different mediums.

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