# Phase Velocity | Group Velocity | Relation between Phase Velocity and Group Velocity for De Broglie Waves

## ▶Phase velocity

When a monochromatic wave travels through a medium, its velocity of advancement in the medium is called * phase velocity*. It is represented by

**v**

_{p}and given by

`v_p=\frac\omega k`

Where, Ï‰ = angular velocity

k = 2Ï€ / Î» (propagation constant)

Thus the ** Phase velocity** is the velocity with which planes of constant phase advance through the medium.

Phase velocity |

## ▶Group velocity

When a group of waves travel in a diffuse medium, the * phase velocities* of its various components vary. Thus, the velocity with which the maximum amplitude of the group advances through the medium is called the

*. It is represented by*

**group velocity****v**

_{g}given by

`v_g=\frac{\triangle\omega}{\triangle k}`

Group velocity |

## ▶Relation between phase velocity and group velocity

Let **u** and **v**_{g} be the phase velocity and group velocity for a non-relativistic free particle of mass m. If (Î») is the de-broglie wavelength and (Î½) is the frequency of the wave, phase velocity is given by

u = Î½.Î»............(1)

According to de broglie hypothesis

Î» = h / mv.........(2)

Total energy of free particle,

`E_k=\frac1 2mv^2`

also,

`E=h\nu`

`\nu=\frac E h=\frac1 2\frac{mv^2}h........(3)`

Substituting (Î») and (f) from equation (2) and (3) into Eq. (1), we get

`u=\frac1 2\frac{mv^2}h.\frac h{mv}`

`u=\frac1 2v_g`

This is the relation between * phase velocity* and

*for a non-relativistic free particle. Thus for a non-relativistic free particle the group velocity (*

**group velocity****v**

_{g}) is twice of the phase velocity (

**u**).

## Difference Between Phase Velocity and Group Velocity

S.No. |
Phase Velocity |
Group Velocity |

1. | In a normal medium, the phase velocity is greater than the group velocity. | In an anomalous medium, the group velocity is greater than the phase velocity. |

2. | Phase velocity is defined for both, the single waves and superimposed waves. | The group velocity is defined only to the superimposed waves. |

3. | The phase velocity is the velocity of the wave with higher frequency. |
The group velocity is the velocity of the wave with lower frequency. |

4. | The Phase velocity refers to the velocity of a monochromatic wave. | Group velocity is the velocity with which the entire group of waves would travel. |