Phase Velocity and Group Velocity - Civil Engineering (CE) PDF Download

Phase velocity and Group velocity Phase velocity:
The velocity with which a wave travels is called Phase velocity or wave velocity. It is denoted by v_p. It is given by

Where c = velocity of light and v = is velocity of the particle. The above equation gives the relationship between the phase velocity and particle velocity. It is clear from the above equation that, Phase velocity is not only greater than the velocity of the particle but also greater than the velocity of light, which can never happen. Therefore phase velocity has no physical meaning in case of matter waves. Thus concept of group velocity was introduced.

Group velocity:
Matter wave is regarded as the resultant of the superposition of large number of component waves all traveling with different velocities. The resultant is in the form of a packet called wave packet or wave group. The velocity with which this wave group travels is called group velocity. The group velocity is represented by v_g.

Expression for group velocity
Consider two waves having same amplitude A, but of slightly different wavelengths and frequencies traveling in the same direction. The two waves are represented by

When these two waves superimpose upon one another, the resultant of the two waves is given by

Therefore

The above equation represents a wave traveling with the velocity ω/κ. But amplitude of the wave is not constant which is given by

The velocity with which variation in the amplitude is transmitted represents the group velocity, and is given by

Relation between Group velocity and phase velocity 
We know that Phase velocity is given by

Group velocity is given by

Therefore equation (1.11) becomes

Equation (1.12) gives the relation between phase velocity and group velocity.

Relation between group velocity and particle velocity
We know that group velocity is given by

Therefore group velocity = particle velocity

Expression for de-Broglie wavelength(From Group velocity)
We know that group velocity is given by

If m is the mass, v is the velocity and V is the potential energy of the particle, then total energy of the particle is given by

This equation is the de-Broglie wave equation derived from group velocity

Properties of Matter waves
Following are some of the important properties of matter waves:
1. Matter waves are waves associated with the moving particle
2. They are not electro magnetic in nature
3. Wavelength of the matter wave is given by λ = h mv = h p
4. The amplitude of the matter wave at the given point determines the probability of finding the particle at that point at the given instant of time.
5. There is no meaning for Phase velocity in case of matter waves. Only group velocity has meaning.