Relation Between Group Velocity And Phase Velocity

Waves can be in the group and such groups are called as wave packets, so the velocity with a wave packet travels is called group velocity. 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 wave number.

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 direct dependence of group velocity on phase velocity and vise-versa.

Relation Between Group Velocity And Phase Velocity Equation

For the amplitude of wave packet let-

  • ?? is the angular velocity given by ??=2????
  • 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 an progressive wave.

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