 # 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

The phase velocity of a wave is given by the following equation:

$$v_p=\frac{\omega }{k}$$…..(eqn 1)

Rewritting the above equation, we get:

$$\omega=kv_p$$…..(eqn 2)

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

$$\frac{dw}{dk}=v_p+k\frac{dv_p}{dk}$$…..(eqn 3)

As $$v_g=\frac{dw}{dk}$$, (eqn 3) reduces to:

$$v_g=v_p+k\frac{dv_p}{dk}$$

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.

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