Waves can be in a group and such groups are called wave packets, so the velocity with which a wave packet travels is called group velocity. The velocity with which the phase of a wave travels is called phase velocity. The relation between group velocity and phase velocity is proportionate.
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Group Velocity And Phase Velocity
The Group Velocity and Phase Velocity relation can be mathematically written as-
\(\begin{array}{l}V_{g}=V_{p}+k\frac{dV_{p}}{dk}\end{array} \) |
Where,
- Vg is the group velocity.
- Vp is the phase velocity.
- k is the angular wavenumber.
Group Velocity and Phase Velocity relation for Dispersive wave non-dispersive wave
| Type of wave | Condition | Formula |
| Dispersive wave | \(\begin{array}{l}\frac{dV_{p}}{dk}\neq 0\end{array} \) |
\(\begin{array}{l}V_{p}\neq V_{g}\end{array} \) |
| Non-dispersive wave | \(\begin{array}{l}\frac{dV_{p}}{dk}=0\end{array} \) |
\(\begin{array}{l}V_{p}= V_{g}\end{array} \) |
Group Velocity And Phase Velocity Relation
The group velocity is directly proportional to phase velocity. This means-
- When group velocity increases, proportionately phase velocity will also increase.
- When phase velocity increases, proportionately group velocity will also increase.
Thus, we see the direct dependence of group velocity on phase velocity and vice-versa.
Relation Between Group Velocity And Phase Velocity Equation
For the amplitude of wave packet let-
- ω is the angular velocity given by ω=2πf
- k is the angular wave number given by \(\begin{array}{l}k=\frac{2\pi }{\lambda }\end{array} \)
- t is time
- x be the position
- Vp phase velocity
- Vg be the group velocity
The phase velocity of a wave is given by the following equation:
Rewriting the above equation, we get:
Differentiating (eqn 2) w.r.t k we obtain,
As
The above equation signifies the relationship between the phase velocity and the group velocity.
Hope you understood the relation between group velocity and phase velocity of a progressive wave.
Physics Related Topics:
| Phase angle |
| Relation between amplitude and frequency |
| Electromagnetic damping |
| Propagation constant |
Frequently Asked Questions – FAQs
Velocity with which a wave packet travels is called?
Velocity with which the phase of a wave packet travels is called?
What is the condition to be satisfied for dispersive waves?
What is the condition to be satisfied for non-dispersive waves?
What is the relationship equation between Group velocity and Phase velocity?
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