Relation between group velocity and phase velocity

In summary, the conversation is discussing a homework problem with equations and an attempt at a solution. The person is questioning their initial assumption and realizing they may be missing a term. The expert provides a summary of the conversation and points out that the initial assumption was correct, but there may be a missing term in the solution. The expert also mentions that they will check the work later and provides a formula for dk/dw.
  • #1
unscientific
1,734
13

Homework Statement



k2mnp1.png



Homework Equations





The Attempt at a Solution


2hg8yt1.png


Is my initial assumption wrong?
 
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  • #2
unscientific said:

Homework Statement



k2mnp1.png



Homework Equations





The Attempt at a Solution


2hg8yt1.png


Is my initial assumption wrong?

No, it's right.
 
  • #3
rude man said:
No, it's right.

But I seem to be missing out on a 1/vph term..
 
  • #4
unscientific said:
But I seem to be missing out on a 1/vph term..

You asked if you initial assumption was right. I did not check all your work. Will try later.
 
  • #5
You wrote dk/dw = (2pi/0)dn/dw.

However, k depends on more than w:
dk = ∂k/∂w dw + ∂k/∂n dn

So 1/vg = dk/dw = ∂k/∂w + ∂k/∂n dn/dw
= n/c + (w/c) dn/dw
Etc.
 

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

Group velocity refers to the speed at which the overall shape or envelope of a wave moves, while phase velocity refers to the speed at which the individual crests or troughs of a wave move. In simpler terms, group velocity is the speed of the wave as a whole, while phase velocity is the speed of the individual parts of the wave.

2. How are group velocity and phase velocity related?

Group velocity and phase velocity are related by the dispersion relationship, which describes the relationship between the frequency and wavenumber of a wave. The group velocity is equal to the phase velocity multiplied by the group index, which is a measure of how much the phase velocity changes with respect to the frequency or wavenumber.

3. What is the significance of the ratio between group velocity and phase velocity?

The ratio between group velocity and phase velocity can provide insight into the behavior of a wave. For example, when the group velocity is equal to the phase velocity, the wave is said to be non-dispersive, meaning that all parts of the wave travel at the same speed. When the group velocity is less than the phase velocity, the wave is said to be dispersive, meaning that different parts of the wave travel at different speeds.

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

The properties of the medium, such as its density, elasticity, and viscosity, can affect the group velocity and phase velocity of a wave. In general, the group velocity and phase velocity will be higher in a medium with lower density and higher elasticity. Viscosity, or the resistance to flow, can also affect the wave speed, particularly for waves traveling through fluids.

5. Can the group velocity and phase velocity be equal for all types of waves?

No, the group velocity and phase velocity can only be equal for non-dispersive waves. For dispersive waves, the group velocity and phase velocity will have different values, and the ratio between them will depend on the dispersion relationship of the specific wave. Examples of non-dispersive waves include sound waves in air and electromagnetic waves in a vacuum.

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