Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Notes > Electromagnetics > Phase and Group Velocity
Phase and Group Velocity | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download
Introduction
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.Group Velocity And Phase Velocity
The Group Velocity and Phase Velocity relation can be mathematically written as-
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
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
- 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:
…..(eqn 1)
Rewriting the above equation, we get:
…..(eqn 2)
Differentiating (eqn 2) w.r.t k we obtain,
…..(eqn 3)
As
(eqn 3) reduces to:
The above equation signifies the relationship between the phase velocity and the group velocity.
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