Group Velocity of Non-Dispersive Wave Packet

In summary, the group velocity for a non-dispersive wave packet can be calculated by solving the phase velocity equation, v=w/k, with a given functional form. This is because in non-dispersive waves, the group velocity is equal to the phase velocity. For sound waves, the equation can be differentiated with respect to w to obtain the group velocity in terms of the refractive index.
  • #1
waley

Homework Statement


I know that for a dispersive wave packet, the group velocity equals the phase velocity, which is given by v=w/k. But how do I calculate the group velocity of a non-dispersive wave packet? I'm supposed to be giving an example with any functional form.

Homework Equations


group velocity is defined by (partial derivative of w(k))/(partial derivative of k)

The Attempt at a Solution


Since w is not dependent on k, the partial derivative should equal to zero? But I'm not sure how to go from there.
 
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  • #2
Could you possibly mean a photon wavepacket, i.e. a light pulse? I think you got your definition reversed - nondispersive wavepacket has its group and phase velocities equal but the dispersive one has them different. A non-dispersive lightpulse is that propagating in a non-dispersive medium, a medium whose refractive index does not vary with frequency (or wavelength). In this case if you differentiate both sides of the equation ##k = \omega n(\omega)/c ## you will get the group velocity in terms of the refractive index (as a function of ## \omega##).
 
  • #3
blue_leaf77 said:
Could you possibly mean a photon wavepacket, i.e. a light pulse? I think you got your definition reversed - nondispersive wavepacket has its group and phase velocities equal but the dispersive one has them different. A non-dispersive lightpulse is that propagating in a non-dispersive medium, a medium whose refractive index does not vary with frequency (or wavelength). In this case if you differentiate both sides of the equation ##k = \omega n(\omega)/c ## you will get the group velocity in terms of the refractive index (as a function of ## \omega##).
Ah, sorry I realized after that I indeed got my definitions reversed. And I also forgot to add that I'm asking in terms of sound waves - would your equation still hold?
How exactly would I differentiate both sides - Is the derivative of k supposed to be the group velocity, and my "functional form" would be something of n in terms of w that I differentiate with respect to w?
 
  • #4
Actually, would it be sufficient enough if I say that for the non-dispersive relation, since the group velocity equals the phase velocity, I just solve the phase velocity with the equation v=w/k? So if I have a functional form of, let's say, cos(x), the phase and group velocity would just be cos(x)/k?
 

Related to Group Velocity of Non-Dispersive Wave Packet

1. What is the concept of group velocity?

The group velocity of a wave packet is the speed at which the envelope of the wave packet is moving. It represents the average velocity at which energy is transferred by the wave packet. It is different from the phase velocity, which is the speed at which the individual wave crests are moving.

2. How is group velocity calculated?

The group velocity can be calculated by taking the derivative of the dispersion relation with respect to the wave number. In non-dispersive media, the dispersion relation is linear and the group velocity is equal to the phase velocity. In dispersive media, the group velocity is typically less than the phase velocity.

3. Why is the group velocity important?

The group velocity is important because it determines the speed at which information can be transmitted by a wave packet. It also plays a role in phenomena such as wave interference and the propagation of wave packets through different mediums.

4. What is a non-dispersive wave packet?

A non-dispersive wave packet is a type of wave packet that maintains its shape and amplitude as it propagates through a medium. This means that the group velocity of the wave packet is equal to the phase velocity. Non-dispersive wave packets are often used to transmit information in communication systems.

5. How does the group velocity of a non-dispersive wave packet change in different mediums?

In non-dispersive media, the group velocity remains constant regardless of the medium it is traveling through. However, in dispersive media, the group velocity can change depending on the properties of the medium, such as its refractive index. This is because the dispersion relation, and therefore the group velocity, is influenced by the medium's properties.

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