How Are Group Velocity and Phase Velocity Related in Dispersive vs Non-Dispersive Media?
Waves are a fundamental concept in physics, observed in both classical and modern domains. Two essential parameters describing waves are group velocity and phase velocity. Understanding their relationship is crucial to mastering wave behavior in fields like optics, quantum mechanics, and acoustics.
Phase velocity is the rate at which the phase of a single-frequency wave propagates through space. This represents the speed at which a fixed phase point (such as a crest) travels. For a sinusoidal wave, phase velocity (Vp) is defined by:
Vp = λ / T
Here, λ is the wavelength and T is the time period.
Alternatively, using angular frequency (ω) and wave number (k):
Vp = ω / k
Group velocity is the speed at which the envelope or the overall shape of a group of waves (wave packet) propagates. It tells how fast energy or information carried by the wave packet travels through the medium. Group velocity (Vg) is given by:
Vg = dω / dk
Where dω/dk denotes the derivative of angular frequency with respect to wave number.
Let's understand the derivation of the relation between group velocity and phase velocity step by step. This is important for conceptual clarity and numerical problem-solving.
Derivation: Relation Between Group Velocity and Phase Velocity
Start with the formula for phase velocity:
Since group velocity Vg = dω/dk:
This equation expresses how the group velocity is connected to phase velocity and how phase velocity changes with wave number.
Table: Key Formulas and Variables
| Physical Quantity | Symbol | Formula | Unit |
|---|---|---|---|
| Phase Velocity | Vp | ω / k | m/s |
| Group Velocity | Vg | dω/dk | m/s |
| General Relation | - | Vg = Vp + k (dVp/dk) | m/s |
Visualizing Dispersive and Non-Dispersive Media
| Media Type | Relation | Condition |
|---|---|---|
| Dispersive | Vg ≠ Vp | dVp/dk ≠ 0 |
| Non-Dispersive | Vg = Vp | dVp/dk = 0 |
Stepwise Problem-Solving Approach
- Identify wave properties: establish ω(k) from the question.
- Calculate phase velocity: Vp = ω / k.
- Find group velocity: Vg = dω/dk.
- Apply Vg = Vp + k (dVp/dk) if needed.
- Check if the wave is dispersive (Vg ≠ Vp) or non-dispersive (Vg = Vp).
Solved Example
Example: Consider ω = A k + B k3, where A and B are constants. Find phase velocity and group velocity.
Group velocity: Vg = dω/dk = A + 3B k2
Difference Between Group Velocity and Phase Velocity
| Feature | Group Velocity (Vg) | Phase Velocity (Vp) |
|---|---|---|
| Definition | Velocity of wave group or envelope | Velocity of a single wave phase/error |
| Formula | dω/dk | ω/k |
| Role in Energy Transfer | Indicates energy/information speed | Does not transfer energy |
| Equality in Media | Equals phase velocity in non-dispersive media | Equals group velocity in non-dispersive media |
Important Facts & Summary
- Both group and phase velocity use angular frequency (ω) and wave number (k).
- Group velocity tells how fast energy moves in the wave packet.
- In dispersive media, phase and group velocities are not equal and depend on wave frequency.
- In non-dispersive media, phase and group velocities are equal.
- Common examples: Pulse in a string (group velocity); light wave (phase velocity).
Practice & Next Steps
- Try more questions using Vedantu's Group and Phase Velocity Practice Materials.
- Revise these concepts before attempting related problems in modern physics and wave mechanics.
- Review solved numericals for better exam performance and concept retention.
Keep practicing the difference and relationship between group velocity and phase velocity to confidently solve problems on waves, optics, and quantum mechanics.
FAQs on Group Velocity vs Phase Velocity: Meaning, Formulas, and Examples
1. What is the relation between group velocity and phase velocity?
The relation between group velocity and phase velocity is given by:
vg = vp + k(dvp/dk).
Here, vg is the group velocity, vp is the phase velocity, k is the wave number, and dvp/dk is the derivative of phase velocity with respect to the wave number. This formula is crucial in understanding how the velocities relate in both dispersive and non-dispersive media.
2. What is phase velocity?
Phase velocity is the speed at which a specific phase of the wave (like the crest) propagates through space.
Formula: vp = ω / k
- ω = angular frequency
- k = wave number
It represents the velocity of a single monochromatic wave component and does not always indicate the speed of energy transfer.
3. What is group velocity?
Group velocity is the speed at which the envelope of a group of waves (wave packet) travels, often corresponding to the velocity of energy or information transfer.
Formula: vg = dω/dk
- ω = angular frequency
- k = wave number
In non-dispersive media, group velocity equals phase velocity, while in dispersive media, they generally differ.
4. When is group velocity equal to phase velocity?
Group velocity equals phase velocity in a non-dispersive medium.
This occurs when the phase velocity does not depend on wave number (k); i.e., dvp/dk = 0. Thus, vg = vp.
5. What is the physical significance of group velocity?
Group velocity represents the speed at which energy or information is transmitted by a wave packet.
- It is the velocity of the envelope of superimposed waves
- Associated with actual particle or signal propagation
- Important in optics, acoustics, and quantum mechanics
6. Why do group velocity and phase velocity differ in a dispersive medium?
Group and phase velocities differ in a dispersive medium because:
- The phase velocity changes with wave number (k)
- Dispersion causes different frequencies to travel at different speeds
- dvp/dk ≠ 0, so vg ≠ vp
7. How do you derive the relation vg = vp + k(dvp/dk)?
To derive the relation:
1. Start with phase velocity: vp = ω / k
2. Rewrite: ω = k vp
3. Differentiate both sides w.r.t k: dω/dk = vp + k(dvp/dk)
4. Therefore, group velocity vg = dω/dk = vp + k(dvp/dk)
8. What is the difference between group velocity and phase velocity?
Main differences:
- Group velocity (vg): Speed of wave packet envelope/energy transfer, formula: dω/dk
- Phase velocity (vp): Speed of single frequency component, formula: ω/k
- In dispersive media: vg ≠ vp. In non-dispersive media: vg = vp
9. Give an example where phase velocity is greater than group velocity.
Example: In matter (de Broglie) waves associated with non-relativistic particles, the phase velocity is typically greater than the group velocity.
For a non-relativistic particle:
- Phase velocity vp = c2 / v
- Group velocity vg = v
Thus, vp > vg
10. What happens to group and phase velocities for electromagnetic waves in vacuum?
For electromagnetic waves in a vacuum:
- The medium is non-dispersive
- All frequencies travel at the same speed (c)
- Group velocity = Phase velocity = c
11. How do you calculate phase velocity and group velocity from a given ω–k relation?
Calculation steps:
1. Identify ω (angular frequency) as a function of k (wave number)
2. Phase velocity: vp = ω / k
3. Group velocity: vg = dω/dk
12. Why is group velocity considered the velocity of energy transfer?
Group velocity is the velocity at which energy and information are transmitted by the wave packet envelope.
- Represents how the modulation (envelope) travels
- In most cases, group velocity matches signal or energy propagation speed
- Crucial for understanding signal transmission in physics and engineering
