Relation Between Group Velocity And Phase Velocity

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.

Table of Contents:

Group Velocity And Phase Velocity

The Group Velocity and Phase Velocity relation can be mathematically written as-

Vg=Vp+kdVpdk

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
dVpdk0
VpVg
Non-dispersive wave
dVpdk=0
Vp=Vg

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
    k=2πλ
  • 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:

vp=ωk
…..(eqn 1)

Rewriting the above equation, we get:

ω=kvp
…..(eqn 2)

Differentiating (eqn 2) w.r.t k we obtain,

dwdk=vp+kdvpdk
…..(eqn 3)


As
vg=dwdk
(eqn 3) reduces to:


vg=vp+kdvpdk

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

Q1

Velocity with which a wave packet travels is called?

Group Velocity is the velocity with which a wave packet travels.
Q2

Velocity with which the phase of a wave packet travels is called?

Phase Velocity is the velocity with which the phase of a wave packet travels.
Q3

What is the condition to be satisfied for dispersive waves?

The condition for dispersive waves is:
dVpdk0
Q4

What is the condition to be satisfied for non-dispersive waves?

The condition for non-dispersive waves is:
dVpdk=0
Q5

What is the relationship equation between Group velocity and Phase velocity?

The equation is:
vg=vp+kdVpdk
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