# Relation Between Group Velocity And Phase Velocity

Waves can be in the group and such groups are called wave packets, so the velocity with 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 are proportionate.

## Group Velocity And Phase Velocity

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

 $V_{g}=V_{p}+k\frac{dV_{p}}{dk}$

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 $\frac{dV_{p}}{dk}\neq 0$ $V_{p}\neq V_{g}$ Non-dispersive wave $\frac{dV_{p}}{dk}=0$ $V_{p}= V_{g}$

### Group Velocity And Phase Velocity Relation

The group velocity is directly proportional to phase velocity. Which 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 – $k=\frac{2\pi }{\lambda }$
• t is time
• x be the position
• Vp phase velocity
• Vg be the group velocity

For any propagating wave packet –

$\frac{\Delta \omega }{2}t-\frac{\Delta k}{2}x=constant$ $\Rightarrow x=constant+\frac{\frac{\Delta \omega }{2}}{\frac{\Delta k}{2}}t$—–(1)

Velocity is the rate of change of displacement given by

$v=\frac{dx}{dt}$

Hence, group velocity is got by differentiating equation (1) with respect to time

$V_{g}=\frac{dx}{dt}=\frac{\frac{\Delta \omega }{2}}{\frac{\Delta k}{2}}=\frac{\Delta \omega }{\Delta k}$ $V_{g}=\lim_{\omega _{1}\rightarrow\omega _{2}} \frac{\Delta \omega }{\Delta k}=\frac{d\omega }{dk}$ ——(2)

We know that phase velocity is given by $V_{g}=\frac{\omega }{k}\;\;\;\Rightarrow \omega =kV_{p}$

Substituting ?? = kVp in equation (2) we get-

$V_{g}=\frac{d\left ( kV_{p} \right )}{dk}$

Thus, we arrive at the equation relating group velocity and phase velocity –

$V_{g}=V_{p}+k\frac{dV_{p}}{dk}$

Hope you understood the relation between group velocity and phase velocity of a progressive wave.

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